http://www.rga78.com/blog
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2015-10-15
http://www.rga78.com/blog/2015/9/23/intro-to-regression-part-8-multiple-regression-regressing-on-two-numeric-variables
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2015-09-23
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Blog - Intro to Regression: Part 8: Multiple regression: regressing on two numeric variables
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Blog - Intro to Regression: Part 8: Multiple regression: regressing on two numeric variables
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Blog - Intro to Regression: Part 8: Multiple regression: regressing on two numeric variables
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Blog - Intro to Regression: Part 8: Multiple regression: regressing on two numeric variables
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Blog - Intro to Regression: Part 8: Multiple regression: regressing on two numeric variables
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Blog - Intro to Regression: Part 8: Multiple regression: regressing on two numeric variables
http://www.rga78.com/blog/2015/9/7/intro-to-regression-part-7-multiple-regression-combining-numeric-and-factor-predictor-variables
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2015-09-21
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Blog - Intro to Regression: Part 7: Multiple regression: combining numeric and factor predictor variables
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Blog - Intro to Regression: Part 7: Multiple regression: combining numeric and factor predictor variables
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Blog - Intro to Regression: Part 7: Multiple regression: combining numeric and factor predictor variables
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Blog - Intro to Regression: Part 7: Multiple regression: combining numeric and factor predictor variables
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Blog - Intro to Regression: Part 7: Multiple regression: combining numeric and factor predictor variables
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Blog - Intro to Regression: Part 7: Multiple regression: combining numeric and factor predictor variables
http://www.rga78.com/blog/2015/8/26/intro-to-regression-part-6-regressing-against-a-factor-variable
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2015-09-07
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Blog - Intro to Regression: Part 6: Regressing against a factor variable
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Blog - Intro to Regression: Part 6: Regressing against a factor variable
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Blog - Intro to Regression: Part 6: Regressing against a factor variable
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Blog - Intro to Regression: Part 6: Regressing against a factor variable
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Blog - Intro to Regression: Part 6: Regressing against a factor variable
http://www.rga78.com/blog/2015/8/25/intro-to-regression-part-5-interpretting-coefficients-centering-predictor-variables
monthly
0.5
2015-09-07
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Blog - Intro to Regression: Part 5: Interpreting coefficients, centering predictor variables
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Blog - Intro to Regression: Part 5: Interpreting coefficients, centering predictor variables
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Blog - Intro to Regression: Part 5: Interpreting coefficients, centering predictor variables
http://www.rga78.com/blog/2015/8/24/intro-to-regression-part-4-distribution-of-prediction-errors-residuals-and-goodness-of-fit-rsup2sup
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0.5
2015-09-07
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Blog - Intro to Regression: Part 4: Distribution of prediction errors (residuals) and goodness of fit (R<sup>2</sup>)
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Blog - Intro to Regression: Part 4: Distribution of prediction errors (residuals) and goodness of fit (R<sup>2</sup>)
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Blog - Intro to Regression: Part 4: Distribution of prediction errors (residuals) and goodness of fit (R<sup>2</sup>)
http://www.rga78.com/blog/2015/8/23/intro-to-regression-part-3-covariance-and-correlation
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0.5
2015-09-23
http://www.rga78.com/blog/2015/8/23/intro-to-regression-part-2-simple-linear-regression-an-example
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2015-09-07
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Blog - Intro to Regression: Part 2: Simple linear regression, an example
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Blog - Intro to Regression: Part 2: Simple linear regression, an example
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Blog - Intro to Regression: Part 2: Simple linear regression, an example
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Blog - Intro to Regression: Part 2: Simple linear regression, an example
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Blog - Intro to Regression: Part 2: Simple linear regression, an example
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Blog - Intro to Regression: Part 2: Simple linear regression, an example
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Blog - Intro to Regression: Part 2: Simple linear regression, an example
http://www.rga78.com/blog/2015/8/20/intro-to-regression-part-1-what-is-regression-generally-speaking
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2015-08-27
http://www.rga78.com/blog/2015/7/20/intro-to-statistics-part-19-confidence-intervals
monthly
0.5
2015-10-15
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Blog - Intro to Statistics: Part 19: Confidence Intervals
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Blog - Intro to Statistics: Part 19: Confidence Intervals
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Blog - Intro to Statistics: Part 19: Confidence Intervals
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Blog - Intro to Statistics: Part 19: Confidence Intervals
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Blog - Intro to Statistics: Part 19: Confidence Intervals
http://www.rga78.com/blog/2015/7/12/intro-to-statistics-part-18-statistical-power
monthly
0.5
2015-08-26
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Blog - Intro to Statistics: Part 18: Type I Errors, Type II Errors, and Statistical Power
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Blog - Intro to Statistics: Part 18: Type I Errors, Type II Errors, and Statistical Power
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Blog - Intro to Statistics: Part 18: Type I Errors, Type II Errors, and Statistical Power
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Blog - Intro to Statistics: Part 18: Type I Errors, Type II Errors, and Statistical Power
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Blog - Intro to Statistics: Part 18: Type I Errors, Type II Errors, and Statistical Power
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Blog - Intro to Statistics: Part 18: Type I Errors, Type II Errors, and Statistical Power
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Blog - Intro to Statistics: Part 18: Type I Errors, Type II Errors, and Statistical Power
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Blog - Intro to Statistics: Part 18: Type I Errors, Type II Errors, and Statistical Power
http://www.rga78.com/blog/2015/6/3/using-apache-nutchsolr-to-build-a-search-engine-with-auto-complete-feature
monthly
0.5
2016-10-10
http://www.rga78.com/blog/2015/6/30/intro-to-statistics-part-17-f-test-anova-significance-testing-between-three-or-more-samples
monthly
0.5
2015-08-20
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
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Blog - Intro to Statistics: Part 17: F-test (ANOVA) Significance Testing Between Three or more Samples
Source: wikipedia
http://www.rga78.com/blog/2015/6/22/intro-to-statistics-part-16-significance-testing-between-two-samples-t-tests
monthly
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2015-08-18
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Blog - Intro to Statistics: Part 16: t-test Significance Testing Between Two Samples
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Blog - Intro to Statistics: Part 16: t-test Significance Testing Between Two Samples
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Blog - Intro to Statistics: Part 16: t-test Significance Testing Between Two Samples
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Blog - Intro to Statistics: Part 16: t-test Significance Testing Between Two Samples
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Blog - Intro to Statistics: Part 16: t-test Significance Testing Between Two Samples
http://www.rga78.com/blog/2015/6/1/intro-to-statistics-part-12-t-tests-and-the-t-distribution
monthly
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2015-08-18
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Blog - Intro to Statistics: Part 15: The t-distribution
\begin{align*} Z & = \frac{\overline{X} - \mu}{ \frac{\sigma}{\sqrt{N}} } \\ \\ where, \\[8pt] \overline{X} & = the \ sample \ mean \\[8pt] \mu & = the \ population \ mean \\[8pt] \sigma & = the \ population \ standard \ deviation \\[8pt] N & = the \ sample \ size \end{align*}
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Blog - Intro to Statistics: Part 15: The t-distribution
\mathchardef\mhyphen="2D \begin{align*} z\mhyphen statistic & = \frac{\overline{X} - \mu}{ \frac{\sigma}{\sqrt{N}} } \\ \\ t\mhyphen statistic & = \frac{\overline{X} - \mu}{ \frac{s}{\sqrt{N}} } \\ \\ where, \\[8pt] \overline{X} & = the \ sample \ mean \\[8pt] \mu & = the \ population \ mean \\[8pt] \sigma & = the \ population \ standard \ deviation \\[8pt] s & = the \ unbiased \ sample \ standard \ deviation \\[8pt] N & = the \ sample \ size \end{align*}
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Blog - Intro to Statistics: Part 15: The t-distribution
\begin{align*} s^2 & = \frac{N}{N-1} \cdot \sigma_x^2 \\[8pt] s^2 & = \frac{30}{30-1} \cdot 9 \\[8pt] s^2 & = 9.3 \\[8pt] s & = \sqrt{9.3} = 3.1 \\ \\ where, \\[8pt] \sigma_x^2 & = biased \ sample \ variance \\[8pt] s^2 & = unbiased \ sample \ variance \\[8pt] s & = unbiased \ sample \ standard \ deviation \\[8pt] N & = the \ sample \ size \end{align*}
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Blog - Intro to Statistics: Part 15: The t-distribution
\mathchardef\mhyphen="2D \begin{align*} t\mhyphen statistic & = \frac{\overline{X} - \mu}{ \frac{s}{\sqrt{N}} } \\[8pt] & = \frac{68.5 - 67}{ \frac{3.1}{\sqrt{30} } } \\[8pt] & = 2.650 \end{align*}
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Blog - Intro to Statistics: Part 15: The t-distribution
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Blog - Intro to Statistics: Part 15: The t-distribution
source: wikipedia
http://www.rga78.com/blog/2015/6/10/intro-to-statistics-part-14-chi-squared-distribution
monthly
0.5
2015-08-17
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
\begin{align*} Q & = \sum_{i=1}^k X_i^2 \\[8pt] & = X_1^2 + X_2^2 + \ ... \ + X_k^2 \\ \\ Q & \sim \chi_k^2 \\ \\ where, \\[8pt] X_1&, \ X_2, \ ... \ X_k \ are \ k \ independent \ standard \ normal \ random \ variables \\[8pt] Q & = chi-squared \ random \ variable \\[8pt] \chi_k^2& \ denotes \ the \ chi-squared \ distribution \ with \ degrees \ of \ freedom = k
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
source: Wikipedia
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
\begin{align*} \operatorname{E}[\chi_k^2] & = k \\ \\ where, \\[8pt] k & = degrees \ of \ freedom \end{align*}
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
(n-1) \frac{s^2}{\sigma^2} \sim \chi_{n-1}^2
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
\begin{align*} s^2 &= \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X})^2 \\[8pt] (n-1) \cdot s^2 &= \sum_{i=1}^n (X_i - \overline{X})^2 \end{align*}
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
\begin{align*} \operatorname{E}[\overline{X}] &= \mu = 0 \\[8pt] (n-1) \cdot s^2 &= \sum_{i=1}^n (X_i - 0)^2 \\[8pt] (n-1) \cdot s^2 &= \sum_{i=1}^n X_i^2 \end{align*}
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Blog - Intro to Statistics: Part 14: The Chi-Squared Distribution
http://www.rga78.com/blog/2015/6/7/intro-to-statistics-part-13-population-variance-vs-sample-variance
monthly
0.5
2015-08-17
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Blog - Intro to Statistics: Part 13: Estimating Population Variance from Sample Variance
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Blog - Intro to Statistics: Part 13: Estimating Population Variance from Sample Variance
\begin{align*} \overline{X} & = \frac{1}{n} \sum_{i=1}^n X_i \\[8pt] \sigma_x^2 & = \frac{1}{n} \sum_{i=1}^n (X_i- \overline{X})^2 \\ \\ where, & \\[8pt] n & = the \ sample \ size \\[8pt] X_1&,\ X_2,\ ...\ X_n \ are \ the \ observations \ in \ the \ sample \\[8pt] \overline{X}& = the \ sample \ mean \\[8pt] \sigma_x^2& = the \ sample \ variance \end{align*}
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Blog - Intro to Statistics: Part 13: Estimating Population Variance from Sample Variance
\begin{align*} \operatorname{E}[\sigma_x^2] & = \frac{n-1}{n} \cdot \sigma^2 \\ \\ where, & \\[8pt] \sigma_x^2& = the \ sample \ variance \\[8pt] \sigma^2& = the \ true \ population \ variance \end{align*}
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Blog - Intro to Statistics: Part 13: Estimating Population Variance from Sample Variance
\begin{align*} s^2 & = \frac{n}{n-1} \cdot \sigma_x^2 \\[8pt] & = \frac{n}{n-1} \cdot \frac{1}{n} \sum_{i=1}^n (X_i- \overline{X})^2 \\[8pt] & = \frac{1}{n-1} \sum_{i=1}^n (X_i- \overline{X})^2 \\ \\ where, & \\[8pt] \sigma_x^2 & = the \ biased \ sample \ variance \\[8pt] s^2 & = the \ unbiased \ sample \ variance \end{align*}
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Blog - Intro to Statistics: Part 13: Estimating Population Variance from Sample Variance
http://www.rga78.com/blog/2015/6/3/intro-to-statistics-part-xx-statistical-significance-testing-using-z-scores
monthly
0.5
2015-08-17
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Blog - Intro to Statistics: Part 12: Statistical Significance Testing Using Z-Scores
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Blog - Intro to Statistics: Part 12: Statistical Significance Testing Using Z-Scores
\begin{align*} \overline{X} & \sim \operatorname{N}\left(\mu, \frac{\sigma^2}{N}\right ) \\[8pt] \operatorname{E}[\overline{X}] & = \mu = 67in \\[8pt] \operatorname{Var}(\overline{X}) & = \frac{\sigma^2}{N} = \frac{3^2}{30} \\[8pt] \operatorname{SE}_{\overline{X}} & = \sqrt{\frac{\sigma^2}{N}} = \frac{\sigma}{\sqrt{N}} = \frac{3}{\sqrt{30}} \\ \\ where, \\[8pt] \overline{X} & = \ the \ sample \ mean \end{align*}
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Blog - Intro to Statistics: Part 12: Statistical Significance Testing Using Z-Scores
\begin{align*} Z & = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{N}}} \\[8pt] Z & = \frac{68.5 - 67}{\frac{3}{\sqrt{30}}} \\[8pt] & = 2.739 \end{align*}
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Blog - Intro to Statistics: Part 12: Statistical Significance Testing Using Z-Scores
http://www.rga78.com/blog/2015/5/30/surfapicom-one-stop-shopping-for-javadoc-api
monthly
0.5
2016-10-10
http://www.rga78.com/blog/2015/5/25/intro-to-statistics-part-11-statistical-significance-and-null-hypothesis-testing
monthly
0.5
2015-08-17
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Blog - Intro to Statistics: Part 11: Statistical Significance and Null Hypothesis Testing
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Blog - Intro to Statistics: Part 11: Statistical Significance and Null Hypothesis Testing
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Blog - Intro to Statistics: Part 11: Statistical Significance and Null Hypothesis Testing
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Blog - Intro to Statistics: Part 11: Statistical Significance and Null Hypothesis Testing
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Blog - Intro to Statistics: Part 11: Statistical Significance and Null Hypothesis Testing
http://www.rga78.com/blog/2015/5/18/intro-to-statistics-part-10-z-scores-standardizing
monthly
0.5
2015-08-14
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
\begin{align*} z & = \frac{x-\mu}{\sigma} \\ \\ where, \\ x & \ is\ the\ outcome \\ \mu& \ is\ the\ mean \\ \sigma& \ is\ the\ standard\ deviation \end{align*}
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
\begin{align*} z_h & = \frac{x-\mu_h}{\sigma_h} \\[8pt] z_h & = \frac{74-67}{4} \\[8pt] z_h & = 1.75 \end{align*}
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
\begin{align*} z_w & = \frac{x-\mu_w}{\sigma_w} \\[8pt] z_w & = \frac{89-70}{10} \\[8pt] z_w & = 1.9 \end{align*}
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
\begin{align*} generic\ normal\ distribution & = \operatorname{N}(\mu,\sigma^2) \\[8pt] standard\ normal\ distribution & = \operatorname{N}(0,1) \end{align*}
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
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Blog - Intro to Statistics: Part 10: Z-scores, Standardizing, and the Standard Normal Distribution
http://www.rga78.com/blog/2015/5/11/intro-to-statistics-part-9-the-central-limit-theorem
monthly
0.5
2015-08-14
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Blog - Intro to Statistics: Part 9: The Central Limit Theorem
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Blog - Intro to Statistics: Part 9: The Central Limit Theorem
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Blog - Intro to Statistics: Part 9: The Central Limit Theorem
\begin{align*} 1. & \quad \quad \overline{X} \sim \operatorname{N}\left(\mu, \frac{\sigma^2}{N}\right ) \\[8pt] 2. & \quad \quad \operatorname{E}[\overline{X}] = \mu = \operatorname{E}[X] \\[8pt] 3. & \quad \quad \operatorname{Var}(\overline{X}) = \frac{\sigma^2}{N} \\[8pt] 4. & \quad \quad \operatorname{SE}_{\overline{X}} = \sqrt{\frac{\sigma^2}{N}} = \frac{\sigma}{\sqrt{N}} \\ \\ where, \\ \overline{X} & \ is \ the \ sample \ mean \ (a \ random \ variable \ itself) \\[8pt] \operatorname{N}&\left(\mu, \frac{\sigma^2}{N}\right ) \ is \ notation \ for, \ normally \ distributed \ with \ mean=\mu \ and\ variance= \frac{\sigma^2}{N} \\[8pt] \mu & = \operatorname{E}[X] \\[8pt] \sigma^2 & = \operatorname{Var}(X) \\[8pt] \sigma & = \sqrt{\operatorname{Var}(X)} \end{align*}
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Blog - Intro to Statistics: Part 9: The Central Limit Theorem
\operatorname{Var}(X) = \frac{n^2-1}{12}
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Blog - Intro to Statistics: Part 9: The Central Limit Theorem
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Blog - Intro to Statistics: Part 9: The Central Limit Theorem
http://www.rga78.com/blog/2015/5/4/sampling-distributions
monthly
0.5
2015-08-13
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Blog - Intro to Statistics: Part 8: Sampling Distributions
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Blog - Intro to Statistics: Part 8: Sampling Distributions
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Blog - Intro to Statistics: Part 8: Sampling Distributions
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Blog - Intro to Statistics: Part 8: Sampling Distributions
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Blog - Intro to Statistics: Part 8: Sampling Distributions
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Blog - Intro to Statistics: Part 8: Sampling Distributions
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Blog - Intro to Statistics: Part 8: Sampling Distributions
http://www.rga78.com/blog/2015/5/1/intro-to-statistics-part-7-common-distributions
monthly
0.5
2015-08-12
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
f(x, \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi} } e^{ -\frac{(x-\mu)^2}{2\sigma^2} }
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
http://en.wikipedia.org/wiki/Normal_distribution
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
f(k;p) = p^k (1-p)^{1-k}\!\quad \text{for }k\in\{0,1\}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
\begin{align*} f(k=0,p) & = p^0 \cdot (1-p)^{1-0} \\[8pt] & = 1 \cdot (1-p)^1 \\[8pt] &= (1-p) \end{align*}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
\begin{align*} f(k=1,p) & = p^1 \cdot (1-p)^{1-1} \\[8pt] & = p \cdot (1-p)^0 \\[8pt] &= p \end{align*}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
\begin{align*} \operatorname{E}[X] & = \sum_{i=1}^{k}x_i\cdot p_i \\[8pt] & = 0 \cdot (1-p) + 1 \cdot p \\[8pt] & = p \end{align*}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
\begin{align*} \operatorname{Var}(X) & = \sum_{i=1}^k p_i\cdot(x_i - \mu)^2 \\[8pt] & = (1-p)\cdot (0 - \operatorname{E}[X])^2 + p\cdot (1 - \operatorname{E}[X])^2 \\[8pt] & = (1-p)\cdot (0 - p)^2 + p\cdot (1 - p)^2 \\[8pt] & = p^2 - p^3 + p - 2p^2 + p^3 \\[8pt] & = p - p^2 \\[8pt] & = p \cdot (1-p) \end{align*}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
f(k;n,p) = {n\choose k}p^k(1-p)^{n-k}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
http://en.wikipedia.org/wiki/Binomial_distribution
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
f(x)=\begin{cases} \frac{1}{b - a} & \mathrm{for}\ a \le x \le b, \\[8] 0 & \mathrm{for}\ x<a\ \mathrm{or}\ x>b \end{cases}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
f(x)=\begin{cases} \frac{1}{b - a + 1} & \mathrm{for}\ a \le x \le b, \\[8] 0 & \mathrm{for}\ x<a\ \mathrm{or}\ x>b \end{cases}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
source: wikipedia
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
source: wikipedia
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
\operatorname{E}[X] = \frac{(b + a)}{2}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
\begin{align*} \operatorname{Var}(X) & = \frac{(b-a)^2}{12} \quad & (continuous) \\ \\ \operatorname{Var}(X) & = \frac{n^2-1}{12} \quad & (discrete) \end{align*}
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
http://en.wikipedia.org/wiki/Poisson_distribution
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Blog - Intro to Statistics: Part 7: Common Distribution Patterns
\begin{align*} \operatorname{E}[X] = \lambda \\ \\ \operatorname{Var}(X) = \lambda \end{align*}
http://www.rga78.com/blog/2015/4/29/intro-to-statistics-part-6-probability-density-functions
monthly
0.5
2015-08-12
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Blog - Intro to Statistics: Part 6: Probability Density Functions
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Blog - Intro to Statistics: Part 6: Probability Density Functions
http://www.rga78.com/blog/2015/4/22/intro-to-statistics-experiments-populations-samples-statistical-inference-oh-my
monthly
0.5
2015-08-12
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
probability\ density = \frac{\{\frac{\#\ of\ observations\ in\ bin}{total\ \#\ of\ observations}\}}{bin\ width}
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
\begin{align*} density & = \frac{\{\frac{150}{1078}\}}{1} \\ \\ & = 0.139 \end{align}
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
\begin{align*} density & = \frac{\{\frac{80}{1078}\}}{0.5} \\ \\ & = 0.148 \end{align}
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
\begin{align*} probability &= 0.148 \times 0.5 \\ \\ &=0.074 \end{align*}
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Blog - Intro to Statistics: Part 5: A Brief Intro to Experiments, Samples, and Statistical Inference
http://www.rga78.com/blog/2015/4/21/an-example-random-variable-rolling-two-dice
monthly
0.5
2015-08-11
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Blog - Intro to Statistics: Part 4: Another Example of a Random Variable: Rolling Two Dice
Source: http://www.free-craps.info/images/dice_chart.gif
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Blog - Intro to Statistics: Part 4: Another Example of a Random Variable: Rolling Two Dice
\begin{align*} \operatorname{E}[X] = x_1&p_1 + x_2p_2 + \dotsb + x_kp_k \\ \\ = 2 \cdot & \frac{1}{36} + 3 \cdot \frac{2}{36} + 4 \cdot \frac{3}{36} + 5 \cdot \frac{4}{36} + 6 \cdot \frac{5}{36} + 7 \cdot \frac{6}{36} \\ \\ & + 8 \cdot \frac{5}{36} + 9 \cdot \frac{4}{36} + 10 \cdot \frac{3}{36} + 11 \cdot \frac{2}{36} + 12 \cdot \frac{1}{36} \\ \\ = 7 \end{align*}
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Blog - Intro to Statistics: Part 4: Another Example of a Random Variable: Rolling Two Dice
\begin{align*} \operatorname{Var}(X) = \operatorname{E}&\left[(X - \mu)^2 \right] \\ \\ = (&2-7)^2 \cdot \frac{1}{36} + (3-7)^2 \cdot \frac{2}{36} + (4-7)^2 \cdot \frac{3}{36} + (5-7)^2 \cdot \frac{4}{36} \\ \\ & + (6-7)^2 \cdot \frac{5}{36} + (7-7)^2 \cdot \frac{6}{36} + (8-7)^2 \cdot \frac{5}{36} + (9-7)^2 \cdot \frac{4}{36} \\ \\ & + (10-7)^2 \cdot \frac{3}{36} + (11-7)^2 \cdot \frac{2}{36} + (12-7)^2 \cdot \frac{1}{36} \\ \\ \operatorname{Var}(X) = 5.&83333 \end{align*}
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Blog - Intro to Statistics: Part 4: Another Example of a Random Variable: Rolling Two Dice
\begin{align*} \operatorname{Var}(X) = \operatorname{E}&[X^2] - \operatorname{E}[X]^2 \\ \\ = 2&^2 \cdot \frac{1}{36} + 3^2 \cdot \frac{1}{36} + 4^2 \cdot \frac{1}{36} + 5^2 \cdot \frac{1}{36} + 6^2 \cdot \frac{1}{36} + 7^2 \cdot \frac{1}{36} \\ \\ + &8^2 \cdot \frac{1}{36} + 9^2 \cdot \frac{1}{36} + 10^2\cdot \frac{1}{36} + 11^2 \cdot \frac{1}{36} + 12^2 \cdot \frac{1}{36} - \operatorname{E}[X]^2 \\ \\ = 5&4.83333 - \operatorname{E}[X]^2 \\ \\ = 5&4.83333 - 7^2 \\ \\ \operatorname{Var}(X) = 5&.83333 \end{align*}
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Blog - Intro to Statistics: Part 4: Another Example of a Random Variable: Rolling Two Dice
Source: http://masamiki.com/project/dice.gif
http://www.rga78.com/blog/2015/4/19/intro-to-statistics-part-3-what-is-variance
monthly
0.5
2015-10-09
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
\begin{align*} \operatorname{Var}(X) & = \operatorname{E}\left[(X - \mu)^2 \right] \\ \\ where \\ \mu & = \operatorname{E}[X] \end{align*}
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
\begin{align*} \operatorname{Var}(X) & = \sum_{i=1}^k p_i\cdot(x_i - \mu)^2 \\ \\ & = p_1\cdot (x_1 - \mu)^2 + p_2\cdot (x_2 - \mu)^2 + ... + p_k\cdot (x_k - \mu)^2 \end{align*}
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
\begin{align*} \operatorname{Var}(X) & = \sum_{i=1}^k p_i\cdot(x_i - \mu)^2 \\ \\ & = \sum_{i=1}^k \left[p_i\cdot x_i^2\right] - \mu^2 \\ \\ & = \operatorname{E}[X^2] - \operatorname{E}[X]^2 \end{align*}
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
\begin{align*} \operatorname{E}[X^2] & = \sum_{i=1}^k p_i\cdot x_i^2 \\ \\ & = p_1\cdot x_1^2 + p_2\cdot x_2^2 + \cdots + p_k\cdot x_k^2 \end{align*}
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
\begin{align*} \sigma & = \sqrt{\operatorname{Var}(X)} \\ \\ \sigma^2 & = \operatorname{Var}(X) \end{align*}
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
\begin{align*} \operatorname{Var}(X) & = \operatorname{E}\left[(X - \mu)^2 \right] \\ \\ & = \frac{1}{6}\cdot (1-3.5)^2 + \frac{1}{6}\cdot (2-3.5)^2 + \frac{1}{6}\cdot (3-3.5)^2 + \frac{1}{6}\cdot (4-3.5)^2 + \frac{1}{6}\cdot (5-3.5)^2 + \frac{1}{6}\cdot (6-3.5)^2 \\ \\ & = 2.91667 \end{align*}
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
\begin{align*} \operatorname{Var}(X) & = \operatorname{E}[X^2] - \operatorname{E}[X]^2 \\ \\ & = \frac{1}{6}\cdot 1^2 + \frac{1}{6}\cdot 2^2 + \frac{1}{6}\cdot 3^2 + \frac{1}{6}\cdot 4^2 + \frac{1}{6}\cdot 5^2 + \frac{1}{6}\cdot 6^2 - \operatorname{E}[X]^2 \\ \\ & = 15.1667 - 3.5^2 \\ \\ & = 2.91667 \end{align*}
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Blog - Intro to Statistics: Part 3: A Random Variable's Variance
\begin{align*} \operatorname{Var}(X) & = (0-0.5)^2 \cdot 0.5 + (1-0.5)^2 \cdot 0.5 \\ \\ & = 0.25 \end{align*}
http://www.rga78.com/blog/2015/4/19/intro-to-statistics-part-2-what-is-a-random-variables-distribution-and-expected-value
monthly
0.5
2015-10-09
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Blog - Intro to Statistics: Part 2: A Random Variable's Distribution and Expected Value
\begin{align*} \operatorname{E}[X] & = \sum_{i=1}^{k}x_i\cdot p_i \\ \\ & = x_1p_1 + x_2p_2 + \dotsb + x_kp_k \end{align*}
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Blog - Intro to Statistics: Part 2: A Random Variable's Distribution and Expected Value
\begin{align*} \operatorname{E}[X] & = x_1\cdot \frac{1}{k} + x_2\cdot \frac{1}{k} + \dotsb + x_k\cdot \frac{1}{k} \\ \\ & = \frac{x_1 + x_2 + \dotsb + x_k}{k} \end{align*}
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Blog - Intro to Statistics: Part 2: A Random Variable's Distribution and Expected Value
\operatorname{E}[X]= 0\cdot 0.5 + 1\cdot 0.5 = 0.5
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Blog - Intro to Statistics: Part 2: A Random Variable's Distribution and Expected Value
\operatorname{E}[X]= \frac{0+1}{2} = \frac{1}{2}
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Blog - Intro to Statistics: Part 2: A Random Variable's Distribution and Expected Value
http://www.rga78.com/blog/2015/4/19/intro-to-statistics-part-1-what-is-a-random-variable
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2015-10-09
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Blog - Intro to Statistics: Part 1: What is a Random Variable?
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2015-08-24
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