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6.7: Chapter 6 Formulas

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    Uniform Distribution

    \(f(x)=\frac{1}{b-a}, \text { for } a \leq x \leq b\)

    • \({P}(X \geq x)=\mathrm{P}(X>x)=\left(\frac{1}{b-a}\right) \cdot(b-x)\)
    • \(\mathrm{P}(X \leq x)=\mathrm{P}(X<x)=\left(\frac{1}{b-a}\right) \cdot(x-a) \)
    • \({P}\left(x_{1} \leq X \leq x_{2}\right)=\mathrm{P}\left(x_{1}<X<x_{2}\right)=\left(\frac{1}{b-a}\right) \cdot\left(x_{2}-x_{1}\right)\)

    Exponential Distribution

    \(f(x)=\frac{1}{\mu} e^{\left(-\frac{x}{\mu}\right)}, \text { for } x \geq 0\)

    • \(\mathrm{P}(X \geq x)=\mathrm{P}(X>x)=\mathrm{e}^{-x / \mu}\)
    • \(\mathrm{P}(X \leq x)=\mathrm{P}(X<x)=1-\mathrm{e}^{-x / \mu}\)
    • \(\mathrm{P}\left(x_{1} \leq X \leq x_{2}\right)=\mathrm{P}\left(x_{1}<X<x_{2}\right)=e^{\left(-\frac{x_{1}}{\mu}\right)}-e^{\left(-\frac{x_{2}}{\mu}\right)}\)

    Standard Normal Distribution

    \(\mu=0, \sigma=1\)
    \(z \text {-score: } z=\frac{x-\mu}{\sigma}\)
    \(x=z \sigma+\mu\)

    Central Limit Theorem

    \(Z \text {-score: } z=\frac{\bar{x}-\mu}{\left(\frac{\sigma}{\sqrt{n}}\right)}\)
    Normal Distribution Probabilities:

    P(X ≤ x) = P(X < x)

    clipboard_eea15e863439893cc29059d4614e633a0.png

    Excel: =NORM.DIST(x,µ,σ,true)

    TI-84: normalcdf(-1E99,x,µ,σ)

    P(X ≥ x) = P(X > x)

    clipboard_ecf1fa1aca9eec2ff2c669e07309599a7.png

    Excel: = 1–NORM.DIST(x,µ,σ,true)

    TI-84: normalcdf(x,1E99,µ,σ)

    P(x1 ≤ X ≤ x2) = P(x1 < X < x2) =

    clipboard_e4270804cd88ccc07d665e11234858ae5.png

    Excel: =NORM.DIST(x2,µ,σ,true)-

    NORM.DIST(x1,µ,σ,true)

    TI-84: normalcdf(x1,x2,µ,σ)
    Percentiles for Normal Distribution:

    P(X ≤ x) = P(X < x)

    clipboard_e465ed1f9bd0e88c97984dc186e6620fc.png

    Excel: =NORM.INV(area,µ,σ)

    TI-84: invNorm(area,µ,σ)

    P(X ≥ x) = P(X > x)

    clipboard_e8349de5059f5fcba60f1cd6adc736275.png

    Excel: =NORM.INV(1–area,µ,σ)

    TI-84: invNorm(1–area,µ,σ)

    P(x1 ≤ X ≤ x2) = P(x1 < X < x2) =

    clipboard_e78d7f5296e8e2a2a575a28ef72ba3ce5.png

    Excel: x1 =NORM.INV((1–area)/2,µ,σ)

    x2 =NORM.INV(1–((1–area)/2),µ,σ)

    TI-84: x1 = invNorm((1–area)/2,µ,σ)

    x2 =invNorm(1–((1–area)/2),µ,σ)

    This page titled 6.7: Chapter 6 Formulas was last modified on Sun, 12 Mar 2023 17:55:06 GMT and is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Rachel Webb via source content that was edited to the style and standards of the LibreTexts platform.