5.5: Chapter Formula Review
- Page ID
- 14652
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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 \(\bf{x}\): \(P(X<x)=(x-a)\left(\frac{1}{b-a}\right)\)
Area to the Right of \(\bf{x}\): \(P(X>x)=(b-x)\left(\frac{1}{b-a}\right)\)
Area Between \(\bf{c}\) and \(\bf{d}\): \(P(c<X<d)=(d-c)\left(\frac{1}{b-a}\right)\)
provided c>a and d<b-
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 \(\mu\)
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