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Probability: Difference between revisions
→Common Convolutions
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This is important for tests.<br> | This is important for tests.<br> | ||
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]. | ||
===Poisson Distributions=== | |||
Sum of poission is poisson sum of lambda. | |||
===Normal Distributions=== | ===Normal Distributions=== | ||
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> | 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> | ||
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If <math>X \sim \Gamma(k, \theta)</math> then <math>\lambda X \sim \Gamma(k, c\theta)</math>.<br> | If <math>X \sim \Gamma(k, \theta)</math> then <math>\lambda X \sim \Gamma(k, c\theta)</math>.<br> | ||
If <math>X_1 \sim \Gamma(k_1, \theta)</math> and <math>X_2 \sim \Gamma(k_2, \theta)</math> then <math>X_2 + X_2 \sim \Gamma(k_1 + k_2, \theta)</math>. | If <math>X_1 \sim \Gamma(k_1, \theta)</math> and <math>X_2 \sim \Gamma(k_2, \theta)</math> then <math>X_2 + X_2 \sim \Gamma(k_1 + k_2, \theta)</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> | |||
===T-distribution=== | |||
Ratio of normal and squared-root of Chi-sq distribution yields T-distribution. | |||
=== | ===Chi-Sq Distribution=== | ||
The ratio of two normalized Chi-sq is an F-distributions | |||
===F Distribution=== | |||
Too many. See [https://en.wikipedia.org/wiki/F-distribution the Wikipedia Page]. | |||
Most important are Chi-sq and T distribution | |||