# 5.4: Chapter Formula Review

- Page ID
- 5567

## 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 \(\sigma = \mu\)
- Additionally
- \(P(X > x) = e^{(–mx)}\)
- \(P(a < X < b) = e^{(–ma)} – e^{(–mb)}\)

- Poisson probability: \(P(X=x)=\frac{\mu^{x} e^{-\mu}}{x !}\) with mean and variance of \(\mu\)