Probability: Difference between revisions

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==Common Convolutions==
==Relationships between distributions==
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].

Revision as of 03:12, 5 November 2019

Introductory Probability as taught in Sheldon Ross' book


Relationships between distributions

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