A Few Continous Distributions:

The Normal Distribution

Two Parameters:

  • Mean: $-\infty\lt\mu\lt\infty$
  • Variance: $\sigma^2\gt0$

The Probability Density Function:

\[f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2\sigma^2}(x-\mu)^2}\]

$X\sim N(\mu, \sigma^2)$

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The Exponential Distribution

One parameter:

  • Rate: $\lambda\gt0$

The Probability Density Function:

\[f(x) = \lambda e^{-\lambda x}\\ \text{for $x\gt0$}\]

  • Mean:

    \[\begin{align} \mu=E[X]&=\int_{-\infty}^{infty}xf(x)dx\\ &=\int_{0}^{\infty}x\lambda e^{-\lambda x}dx\\ &=\frac{1}{\lambda} \end{align}\]

  • Variance:

    \[\begin{align} \sigma^2=Var[X]&=E[(X-\mu)^2]\\ &=E[X^2] - (E[x])^2\\ &=\frac{1}{\lambda^2} \end{align}\]

  • Notation

    \[X\sim exp(\text{rate}=\lambda)\\ f(x)=\lambda e^{-\lambda x}\]

    Or

    \[X\sim exp(\text{mean}=\lambda\\ f(x)=\frac{1}{\lambda}e^{-x/\lambda}\]

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The Gamma Distribution

Two parameters:

  • Shape: $\alpha\gt0$
  • Inverse Scale: $\beta\gt0$

The Probability Density Function:

\[f(x)=\frac{1}{\Gamma(\alpha)}\beta^{\alpha}x^{\alpha-1}e^{-\beta x}\\ \text{for $x\gt0$}\]

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The Gamma Function:

\[\Gamma(\alpha)=\int_{0}^{\infty}x^{\alpha-1}e^{-x}dx\]

Properties:

  • $\Gamma(1)=1$

  • $\Gamma(\alpha)=(\alpha-1)\Gamma(\alpha-1)\quad\text{for }\alpha\gt1$

  • $\Gamma(n)=(n-1)!\quad\text{for an integer }n\gt1$

\[X\sim\Gamma(\alpha,\beta)\]

  • Mean:

    \[\begin{align} \mu&=E[X]=\int_{-\infty}^{\infty}xf(x)dx\\ &=\int_{0}^{\infty}x\frac{1}{\Gamma(\alpha)}\beta^\alpha x^{\alpha-1}e^{-\beta x}dx\\ &=\frac{\alpha}{\beta} \end{align}\]

  • Variance:

    \[\begin{align} \sigma^2=Var[X]&=E[(X-\mu)^2]\\ &=E[X^2]-(E[X])^2\\ &=\frac{\alpha}{\beta^2} \end{align}\]

The Chi-Squared Distribution

One Parameter:

  • Degrees of freedom: $n\ge1\quad\text{(n is an integer)}$
\[X\sim\chi^2(n)\\ \text{is defined as}\\ \Gamma\bigg(\frac{n}{2},\frac{1}{2}\bigg)\]

  • Mean:

    \[\mu=E[X]=n\]

  • Variance:

    \[\sigma^2=Var[X]=2n\]

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The t-distribution

Let $Z\sim N(0,1)$ and $W\sim\chi^2(n)$ be independent random variables.

Define

\[T=\frac{Z}{\sqrt{W/n}}\]

Then $T$ has pdf.

\[f(x)=\frac{\Gamma\big(\frac{n+1}{2}\big)}{\sqrt{n\pi}\Gamma\big(\frac{n}{2}\big)} \Bigg(1+\frac{x^2}{n}\Bigg)^{-(n+1)/2}\\ (-\infty\lt x\lt\infty)\]

We write $X\sim t(n)$

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For critical value

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Some facts:

  • $Z\sim N(0,1)\implies Z^2\sim\chi^2(1)$

  • $X_1,X_2,…,X_k$ independent with $X_i\sim\chi^2(n_i)

    \[\sum_{i=1}^{k}X_i\sim\chi^2(n_1+n_2+...+n_k)\]

    In particular, $X_1,X_2,…,X_n\stackrel{iid}{\sim}\chi^2(1)$

    \[\implies\sum_{i=1}^{n}X_i\sim\chi^2(n)\]

  • $X\sim\Gamma(\alpha,\beta)$ and $c\gt0$

    \[\implies cX\sim\Gamma(\alpha,\beta/c)\]

  • $X\sim\Gamma(\alpha,\beta)$

    \[f(x)=\frac{1}{\Gamma(\alpha)}\beta^\alpha x^{\alpha-1}e^{-\beta x}\]

    $\alpha=1\implies X\sim exp(\text{rate}=\beta)$

  • $X_1,X_2,…,X_n\stackrel{iid}{\sim}exp(\text{rate}=\lambda)$

    \[\sum_{i=1}^{n}X_i\sim\Gamma(n,\lambda)\]

  • $X_1,X_2,…,X_n\stackrel{iid}{\sim}\Gamma(\alpha,\beta)$

    \[\implies\sum_{i=1}^{n}X_i\sim\Gamma(n\alpha,\beta)\]

Things we now know

  • $X_1,X_2,…,X_n\stackrel{iid}{\sim}exp(\text{rate}=\lambda)$

    \[\begin{align} \overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i\sim\Gamma(n, n\lambda)\\ \implies 2n\lambda\overline{X}&=\Gamma\bigg(n,\frac{1}{2}\bigg)\\ &=\Gamma\bigg(\frac{2n}{2},\frac{1}{2}\bigg)=\chi^2(2n) \end{align}\]

  • $X_1,X_2,…,X_n\stackrel{iid}{\sim}\Gamma(\alpha,\beta)$

    \[\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i\sim\Gamma(n\alpha,n\beta)\]