6.7 Chapter 6 Formulas

Uniform Distribution $f(x)=\frac{1}{b-a},\text{ for }a\le x\le b$ $P(X\ge x)=P(X>x)=\left(\frac{1}{b-a}\right)·(b-x)$ $P(X\le x)=P(X<x)=\left(\frac{1}{b-a}\right)·(x-a)$ $P(x_1\le X\le x_2)=P(x_1<X<x_2)=\left(\frac{1}{b-a}\right)·(x_2-x_1)$	Exponential Distribution $f(x)=\frac{1}{\mu }{e}^{(-\frac{x}{\mu })},\text{ for }x\ge 0$ $P(X\ge x)=P(X>x)=e^{-x/\mu }$ $P(X\le x)=P(X<x)=1-e^{-x/\mu }$ $P(x_1\le X\le x_2)=P(x_1<X<x_2)=e^{(-\frac{x_1}{\mu })}-e^{(-\frac{x_2}{\mu })}$
Standard Normal Distribution $\mu =0,\sigma =1$ $z\text{-score: }z=\frac{x-\mu }{\sigma }$ $x=z\sigma +\mu$	Central Limit Theorem $Z\text{-score: }z=\frac{\overline{x}-\mu }{\left(\frac{\sigma }{\sqrt{n}}\right)}$
Normal Distribution Probabilities:	P(X ≤ x) = P(X < x) Excel: =NORM.DIST(x,µ,σ,true) TI-84: normalcdf(-1E99,x,µ,σ)
P(X ≥ x) = 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) = 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) Excel: =NORM.INV(area,µ,σ) TI-84: invNorm(area,µ,σ)
P(X ≥ x) = 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) = 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), hosted on LibreTexts (stats.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.