# 5.4: Chapter Formula Review

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## 5.1 Properties of Continuous Probability Density Functions

Probability density function (pdf) $$f(x)$$:

• Cumulative distribution function (cdf): $$P(X \leq x)$$

## 5.2 The Uniform Distribution

$$X \sim U (a, b)$$

The mean is $$\mu=\frac{a+b}{2}$$

The standard deviation is $$\sigma=\sqrt{\frac{(b-a)^{2}}{12}}$$

Probability density function: $$f(x)=\frac{1}{b-a} \text { for } a \leq X \leq b$$

Area to the Left of $$\bf{x}$$: $$P(X<x)> Area to the Right of \(\bf{x}$$: $$P(X>x)=(b-x)\left(\frac{1}{b-a}\right)$$

Area Between $$\bf{c}$$ and $$\bf{d}$$: $$P(c<d)> • ## 5.3 The Exponential Distribution • pdf: \(f(x) = me^{(–mx)}$$ where $$x \geq 0$$ and $$m > 0$$
• cdf: $$P(X \leq x) = 1 – e^{(–mx)}$$
• mean $$\mu = \frac{1}{m}$$
• standard deviation $$\sigma = \mu$$
• $$P(X > x) = e^{(–mx)}$$
• $$P(a < X < b) = e^{(–ma)} – e^{(–mb)}$$
• Poisson probability: $$P(X=x)=\frac{\mu^{x} e^{-\mu}}{x !}$$ with mean and variance of $$\mu$$