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Probability: Difference between revisions
→Normal + Normal
(Created page with "Introductory Probability as taught in [https://www.pearson.com/us/higher-education/program/Ross-First-Course-in-Probability-A-9th-Edition/PGM110742.html Sheldon Ross' book]...") |
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See [https://en.wikipedia.org/wiki/Relationships_among_probability_distributions Relationships among probability distributions]. | See [https://en.wikipedia.org/wiki/Relationships_among_probability_distributions Relationships among probability distributions]. | ||
===Normal + Normal=== | ===Normal + Normal=== | ||
If <math>X_1 \sim N(\mu_1, \sigma_1^2)</math> and <math>X_2 \sim N(\mu_2, \sigma_2^2)</math> then <math>\lambda_1 X_1 + \lambda_2 X_2 \sim N(\lambda_1 \mu_1 + \lambda_2 X_2, \lambda_1^2 \sigma_1^2 + \lambda_2^2 + \sigma_2^2</math> for any <math>\lambda_1, \lambda_2 \in \mathbb{R}</math> | |||
===Gamma + Gamma=== | ===Gamma + Gamma=== | ||
Note exponential distributions are also Gamma distrubitions | Note exponential distributions are also Gamma distrubitions | ||
===Gamma and Beta=== | ===Gamma and Beta=== | ||
If <math>X_1 \sim \Gamma(\alpha, \theta)</math> and <math>X_2 \sim \Gamma(\beta, \theta)</math>, then <math>\frac{X_1}{X_1 + X_2} \sim B(\alpha, \beta)</math> | If <math>X_1 \sim \Gamma(\alpha, \theta)</math> and <math>X_2 \sim \Gamma(\beta, \theta)</math>, then <math>\frac{X_1}{X_1 + X_2} \sim B(\alpha, \beta)</math> | ||