Chapter 5 Formula Review

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<d)> • 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$$
• $$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$$