Probability: Difference between revisions
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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 | |||
Revision as of 03:12, 5 November 2019
Introductory Probability as taught in Sheldon Ross' book
Common Convolutions
This is important for tests.
See Relationships among probability distributions.
Poisson Distributions
Sum of poission is poisson sum of lambda.
Normal Distributions
If \(\displaystyle X_1 \sim N(\mu_1, \sigma_1^2)\) and \(\displaystyle X_2 \sim N(\mu_2, \sigma_2^2)\) then \(\displaystyle \lambda_1 X_1 + \lambda_2 X_2 \sim N(\lambda_1 \mu_1 + \lambda_2 X_2, \lambda_1^2 \sigma_1^2 + \lambda_2^2 + \sigma_2^2)\) for any \(\displaystyle \lambda_1, \lambda_2 \in \mathbb{R}\)
Gamma Distributions
Note exponential distributions are also Gamma distrubitions
If \(\displaystyle X \sim \Gamma(k, \theta)\) then \(\displaystyle \lambda X \sim \Gamma(k, c\theta)\).
If \(\displaystyle X_1 \sim \Gamma(k_1, \theta)\) and \(\displaystyle X_2 \sim \Gamma(k_2, \theta)\) then \(\displaystyle X_2 + X_2 \sim \Gamma(k_1 + k_2, \theta)\).
If \(\displaystyle X_1 \sim \Gamma(\alpha, \theta)\) and \(\displaystyle X_2 \sim \Gamma(\beta, \theta)\), then \(\displaystyle \frac{X_1}{X_1 + X_2} \sim B(\alpha, \beta)\)
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 the Wikipedia Page. Most important are Chi-sq and T distribution