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Probability: Difference between revisions
→Relationships between distributions
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* The sample mean converges to the population mean almost surely. | * The sample mean converges to the population mean almost surely. | ||
==Relationships between distributions== | ==Properties and Relationships between distributions== | ||
This is important for exams.<br> | This is important for exams.<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 | ===Poisson Distribution=== | ||
* If <math>X_i \sim Poisson(\lambda_i)</math> then <math>\sum X_i \sim Poisson(\sum \lambda_i)</math> | * If <math>X_i \sim Poisson(\lambda_i)</math> then <math>\sum X_i \sim Poisson(\sum \lambda_i)</math> | ||
===Normal | ===Normal Distribution=== | ||
* 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> | ||
===Exponential | ===Exponential Distribution=== | ||
* If <math>\epsilon_1, ..., \epsilon_n</math> are exponential distributions then <math>\min\{\epsilon_i\} \sim \exp(\sum \lambda_i)</math> | * If <math>\epsilon_1, ..., \epsilon_n</math> are exponential distributions then <math>\min\{\epsilon_i\} \sim \exp(\sum \lambda_i)</math> | ||
* Note that the maximum is not exponentially distributed | * Note that the maximum is not exponentially distributed | ||
* However, if <math>X_1, ..., X_n \sim \exp(1)</math> then <math>Z_n=n\exp(\max\{\epsilon_i\}) \rightarrow \exp(1)</math> | * However, if <math>X_1, ..., X_n \sim \exp(1)</math> then <math>Z_n=n\exp(\max\{\epsilon_i\}) \rightarrow \exp(1)</math> | ||
===Gamma | ===Gamma Distribution=== | ||
Note exponential distributions are also Gamma distrubitions | Note exponential distributions are also Gamma distrubitions | ||
* 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> | ||