5.5: Chapter Formula Review

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 \(\mu\)