6.7: Chapter 6 Formulas

Last updated
Save as PDF

Page ID: 26657

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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)

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)