# 5.5: Chapter Formula Review

- Page ID
- 5567

## 5.1 Properties of Continuous Probability Density Functions

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

- \(f(x) \geq 0\)
- The total area under the curve \(f(x)\) is one.

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

## 5.2 The Uniform Distribution

\(X = a\) real number between \(a\) and \(b\) (in some instances, \(X\) can take on the values \(a\) and \(b\)). \(a =\) smallest \(X; b =\) largest \(X\)

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

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

- pdf: \(f(x)=\frac{1}{b-a} \text { for } a \leq x \leq b\)
- cdf: \(P(X \leq x)=\frac{x-a}{b-a}\)
- mean \(\mu=\frac{a+b}{2}\)
- standard deviation \(\sigma=\sqrt{\frac{(b-a)^{2}}{12}}\)
- \(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 \(\mu\)