5.7: Formula Review
- Page ID
- 45938
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)5.2 Properties of Continuous Probability Density Functions
Probability density function (pdf) \(f(x)\) :
- \(f(x) \geq 0\)
- The total area under the curve \(f(x)\) is one.
Cumulative distribution function (cdf): \(P(X \leq x\) )
5.3 The Uniform Distribution
\(X=\) a real number between \(a\) and \(b\) (in some instances, \(X\) can take on the values \(a\) and \(b\) ). \(a=\) smallest \(X ; b=\) largest \(X\)
\[
X \sim U(\mathrm{a}, \mathrm{b})
\]
The mean is \(\mu=\frac{a+b}{2}\)
The standard deviation is \(\sigma=\sqrt{\frac{(b-a)^2}{12}}\)
Probability density function: \(f(x)=\frac{1}{b-a}\) for \(a \leq X \leq b\)
Area to the Left of \(x: P(X<x)=(x-a)\left(\frac{1}{b-a}\right)\)
Area to the Right of \(x: P(X>x)=(b-x)\left(\frac{1}{b-a}\right)\)
Area Between \(c\) and \(d\) : \(P(c<x<d)=\left(\right.\) base)(height) \(=(d-c)\left(\frac{1}{b-a}\right)\)
- pdf: \(f(x)=\frac{1}{b-a}\) for \(a \leq x \leq b\)
- cdf: \(P(X \leq x)=\frac{x-a}{b-a}\)
- mean \(\mu=\frac{a+b}{2}\)
- standard deviation \(\sigma=\sqrt{\frac{(b-a)^2}{12}}\)
- \(P(c<X<d)=(d-c)\left(\frac{1}{b-a}\right)\)
5.4 The Exponential Distribution
- pdf: \(f(x)=m e^{(-m x)}\) where \(x \geq 0\) and \(m>0\)
- cdf: \(P(X \leq x)=1-e^{(-m x)}\)
- mean \(\mu=\frac{1}{m}\)
- standard deviation \(\sigma=\mu\)
- Additionally
- \(P(X>x)=e^{(-m x)}\)
- \(P(a<X<b)=e^{(-m a)}-e^{(-m b)}\)
- Poisson probability: \(P(X=x)=\frac{\mu^x e^{-\mu}}{x!}\) with mean and variance of \(\mu\)