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Statistics LibreTexts

5.5: Chapter Formula Review

  • Page ID
    14652
  • 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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