Mostly Harmless StatisticsXYZ Homework Edition

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6.7 Chapter 6 Formulas

Uniform Distribution f(x)=1b−a, for a≤x≤bf(x)=\frac{1}{b-a}, \text { for } a \leq x \leq b P(X≥x)=P(X>x)=(1b−a)·(b−x){P}(X \geq x)=\mathrm{P}(X>x)=\left(\frac{1}{b-a}\right) \cdot(b-x) P(X≤x)=P(X<x)=(1b−a)·(x−a)\mathrm{P}(X \leq x)=\mathrm{P}(X<x)=\left(\frac{1}{b-a}\right) \cdot(x-a) P(x1≤X≤x2)=P(x1<X<x2)=(1b−a)·(x2−x1){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)=1μe(−xμ), for x≥0f(x)=\frac{1}{\mu} e^{\left(-\frac{x}{\mu}\right)}, \text { for } x \geq 0 P(X≥x)=P(X>x)=e−x/μ\mathrm{P}(X \geq x)=\mathrm{P}(X>x)=\mathrm{e}^{-x / \mu} P(X≤x)=P(X<x)=1−e−x/μ\mathrm{P}(X \leq x)=\mathrm{P}(X<x)=1-\mathrm{e}^{-x / \mu} P(x1≤X≤x2)=P(x1<X<x2)=e(−x1μ)−e(−x2μ)\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 μ=0,σ=1\mu=0, \sigma=1
z-score: z=x−μσz \text {-score: } z=\frac{x-\mu}{\sigma}
x=zσ+μx=z \sigma+\mu 		Central Limit Theorem Z-score: z=x¯−μ(σn)Z \text {-score: } z=\frac{\bar{x}-\mu}{\left(\frac{\sigma}{\sqrt{n}}\right)}
Normal Distribution Probabilities: P(X ≤ x) = P(X < x) Bell curve with the left-tail area below a value left of the mean shaded green, illustrating the left-tail probability P(X <= x). Excel: =NORM.DIST(x,µ,σ,true) TI-84: normalcdf(-1E99,x,µ,σ)
P(X ≥ x) = P(X > x) Bell curve with the right-tail area above a value right of the mean shaded green, illustrating the right-tail probability P(X >= x). Excel: = 1–NORM.DIST(x,µ,σ,true) TI-84: normalcdf(x,1E99,µ,σ) P(x1 ≤ X ≤ x2) = P(x1 < X < x2) = Bell curve with the central area between two values on either side of the mean shaded green, illustrating the between probability P(x1 <= X <= x2). 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) Bell curve with the left-tail area shaded green, illustrating the percentile case of finding the x-value with a given area to its left, P(X <= x). Excel: =NORM.INV(area,µ,σ) TI-84: invNorm(area,µ,σ)
P(X ≥ x) = P(X > x) Bell curve with the right-tail area shaded green, illustrating the percentile case of finding the x-value with a given area to its right, P(X >= x). Excel: =NORM.INV(1–area,µ,σ) TI-84: invNorm(1–area,µ,σ) P(x1 ≤ X ≤ x2) = P(x1 < X < x2) = Bell curve with the middle area between two values shaded green, illustrating finding the two x-values x1 and x2 that bound a given central area. 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)

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University, https://mostlyharmlessstat.wixsite.com/webpage), © Rachel Webb, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.

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