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===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> | ||
===Exponential Distributions=== | |||
* 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 | |||
* 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 Distributions=== | ===Gamma Distributions=== |