8.8: Formula Review
- Page ID
- 261812
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)4.3 Mean or Expected Value and Standard Deviation
Mean or Expected Value: \(\mu=\sum_{x \in X} x P(x)\)
Standard Deviation: \(\sigma=\sqrt{\sum_{x \in X}(x-\mu)^2 P(x)}\)
4.4 Binomial Distribution
\(X \sim B(n, p)\) means that the discrete random variable \(X\) has a binomial probability distribution with \(n\) trials and probability of success \(p\).
\(X=\) the number of successes in \(n\) independent trials
\(n=\) the number of independent trials
\(X\) takes on the values \(x=0,1,2,3, \ldots, n\)
\(p=\) the probability of a success for any trial
\(q=\) the probability of a failure for any trial
\(p+q=1 \)
\(q=1-p \)
The mean of \(X\) is \(\mu=n p\). The standard deviation of \(X\) is \(\sigma=\sqrt{n p q}\).
4.5 Geometric Distribution
\(X \sim \mathrm{G}(p)\) means that the discrete random variable \(X\) has a geometric probability distribution with probability of success in a single trial \(p\).
\(X=\) the number of independent trials until the first success
\(X\) takes on the values \(x=1,2,3, \ldots\)
\(p=\) the probability of a success for any trial
\(q=\) the probability of a failure for any trial \(p+q=1\)
\(q=1-p\)
The mean is \(\mu=\frac{1}{p}\).
The standard deviation is \(\sigma=\sqrt{\frac{1-p}{p^2}}=\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}\).
4.6 Hypergeometric Distribution
\(X \sim H(r, b, n)\) means that the discrete random variable \(X\) has a hypergeometric probability distribution with \(r=\) the size of the group of interest (first group), \(b=\) the size of the second group, and \(n=\) the size of the chosen sample.
\(X=\) the number of items from the group of interest that are in the chosen sample, and \(X\) may take on the values \(x= 0,1, \ldots\), up to the size of the group of interest. (The minimum value for \(X\) may be larger than zero in some instances.)
\(n \leq r+b\)
The mean of \(X\) is given by the formula \(\mu=\frac{n r}{r+b}\) and the standard deviation is \(=\sqrt{\frac{r b n(r+b-n)}{(r+b)^2(r+b-1)}}\).
4.7 Poisson Distribution
\(X \sim P(\mu)\) means that \(X\) has a Poisson probability distribution where \(X=\) the number of occurrences in the interval of interest.
\(X\) takes on the values \(x=0,1,2,3, \ldots\)
The mean \(\mu\) is typically given.
The variance is \(\sigma^2=\mu\), and the standard deviation is \(\sigma=\sqrt{\mu}\).
The probability of having exactly \(x\) successes in \(r\) trials is \(P(X=x)=\left(e^{-\mu}\right) \frac{\mu^x}{x!}\).
When \(P(\mu)\) is used to approximate a binomial distribution, \(\mu=n p\) where \(n\) represents the number of independent trials and \(p\) represents the probability of success in a single trial.