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Statistics LibreTexts

5.5: Chapter 5 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<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 \(\sigma = \mu\)
    • 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 \(\mu\)