Common Distribution Equations

Notations

• $N(\mu ,{\sigma }^2)$ : Normal distribution with mean $\mu$ and variance ${\sigma }^2$
• ${\chi }^2(k)$ : Chi-square distribution with $k$ degrees of freedom
• $t(r)$ : t-distribution with $r$ degrees of freedom
• $F(r_1,r_2)$ : F-distribution with $r_1$ and $r_2$ degrees of freedom
• $\bar{X}$ : Sample mean
• $S^2$ : Sample variance
• $\stackrel{D}{\to }$ : Convergence in distribution

Sample Mean and Variance Distributions

If $X_1,X_2,\dots ,X_n\sim N(\mu ,{\sigma }^2)$ independently, then:

$$\begin{array}{rlrl}\bar{X}& =\frac{1}{n}\sum _{i=1}^n{X}_i& & \sim N\left(\mu ,\frac{{\sigma }^2}{n}\right)\\ S^2& =\frac{1}{n-1}\sum _{i=1}^n(X_i-\bar{X}{)}^2& & \sim N\left({\sigma }^2,\frac{2{\sigma }^4}{n-1}\right)\end{array}$$

Normal Distribution Relationships

If $X_1,X_2,\dots ,X_n\sim N(\mu ,{\sigma }^2)$ independently, then:

$$\frac{(n-1)S^2}{{\sigma }^2}\sim {\chi }^2(n-1)$$

If $Z_1,Z_2,\dots ,Z_k\sim N(0,1)$ independently, then:

$$\sum _{i=1}^k{Z}_i^2\sim {\chi }^2(k)$$

If $U\sim N(0,1)$ and $V\sim {\chi }^2(r)$ independently, then:

$$T=\frac{U}{\sqrt{V/r}}\sim t(r)$$

If $U\sim {\chi }^2(r_1)$ and $V\sim {\chi }^2(r_2)$ independently, then:

$$F=\frac{U/r_1}{V/r_2}\sim F(r_1,r_2)$$

If $X_1,X_2,\dots ,X_n\sim N(\mu ,{\sigma }^2)$ independently, then:

$$T=\frac{\bar{X}-\mu }{S/\sqrt{n}}\sim t(n-1)$$

For $X_1,\dots ,X_{n_1}\sim N({\mu }_1,{\sigma }^2)$ and $Y_1,\dots ,Y_{n_2}\sim N({\mu }_2,{\sigma }^2)$ independently:

$$T=\frac{(\bar{X}-\bar{Y})-({\mu }_1-{\mu }_2)}{S_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\sim t(n_1+n_2-2)$$

where $S_p^2=\frac{(n_1-1)S_X^2+(n_2-1)S_Y^2}{n_1+n_2-2}$

For independent samples from $N({\mu }_1,{\sigma }_1^2)$ and $N({\mu }_2,{\sigma }_2^2)$

$$F=\frac{S_1^2/{\sigma }_1^2}{S_2^2/{\sigma }_2^2}\sim F(n_1-1,n_2-1)$$

If $X_1,X_2,\dots ,X_n\sim b(1,p)$ independently, then:

$$Y_n=X_1+X_2+\cdots +X_n\sim b(n,p)$$

Central Limit Theorem Results

If $X_1,X_2,\dots ,X_n$ are i.i.d. with $E[X_i]=\mu$ and $\text{Var}(X_i)={\sigma }^2$, then:

$$\begin{array}{rl}\frac{\bar{X}-\mu }{\sigma /\sqrt{n}}& \stackrel{D}{\to }N(0,1)\\ ⟺\sqrt{n}(\bar{X}-\mu )& \stackrel{D}{\to }N(0,{\sigma }^2)\end{array}$$

Note

Usually $n>30$ is sufficient. More skewed distributions may require larger $n$

Gamma Distribution Relationships

${\chi }^2(k)=\text{Gamma}\left(\frac{k}{2},\frac{1}{2}\right)$

If $X_1,\dots ,X_n\sim \text{Exp}(\lambda )$ independently, then: ${\sum }_{i=1}^n{X}_i\sim \text{Gamma}(n,\lambda )$

Order Statistics

For $X_1,\dots ,X_n\sim \text{Uniform}(0,1)$:

• $X_{(n)}\sim \text{Beta}(n,1)$
• $X_{(1)}\sim \text{Beta}(1,n)$
• $X_{(k)}\sim \text{Beta}(k,n-k+1)$

Transformations

Log-Normal Distribution

If $X\sim N(\mu ,{\sigma }^2)$, then: $Y=e^X\sim \text{LogNormal}(\mu ,{\sigma }^2)$

Box-Muller Transformation

If $U_1,U_2\sim \text{Uniform}(0,1)$ independently, then: $Z_1=\sqrt{-2\ln U_1}\cos (2\pi U_2)\sim N(0,1)$ $Z_2=\sqrt{-2\ln U_1}\sin (2\pi U_2)\sim N(0,1)$