5.5: Chapter 5 Formulas
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 5 Formulas
Mean for a Discrete Probability Distribution
The mean of a discrete probability distribution represents the long-run average or central value of the outcomes if an experiment were repeated many times. It is found by multiplying each possible value by its corresponding probability and summing the results.
\( \mu = \sum x \cdot P(x) \)
Where:
- \( \mu \) is the mean
- \( x \) represents each possible value
- \( P(x) \) is the probability of each value \( x \)
Variance for a Discrete Probability Distribution
The variance of a discrete probability distribution measures how much the values of the distribution differ from the mean. It represents the average of the squared differences between each value and the mean, weighted by their probabilities.
\( \sigma^2 = \sum x^2 \cdot P(x) - \mu^2 \)
Where:
- \( \sigma^2 \) is the variance
- \( x \) is each possible value
- \( P(x) \) is the probability of each value \( x \)
- \( \sum x^2 \cdot P(x) \) is the expected value of the square of \( x \)
- \( \mu \) is the mean (expected value)
Standard Deviation for a Discrete Probability Distribution
The standard deviation of a discrete probability distribution is a measure of the spread of the distribution. It tells how much the values typically deviate from the mean. It is calculated by taking the square root of the variance.
\( \sigma = \sqrt{\sigma^2} = \sqrt{\text{Variance}} \)
Where:
- \( \sigma \) is the standard deviation
- \( \sigma^2 \) is the variance of the distribution
Expected Value
The expected value, denoted as \(E(X)\), is the theoretical average of a random variable in a probability distribution. It represents the long-run average outcome if an experiment were repeated many times. For a discrete random variable, it is calculated by multiplying each possible value by its probability and summing the results.
\( E(X) = \sum x \cdot P(x) \)
Where:
- \( E(X) \) is the expected value
- \( x \) is each possible value of the random variable
- \( P(x) \) is the probability of \( x \)
Probability Formula for the Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. Each trial has the same probability of success. The binomial distribution is used when you want to know the likelihood of getting exactly a certain number of successes in repeated trials.
\( P(x) = \dfrac{n!}{x!(n - x)!} \cdot p^x \cdot (1 - p)^{n - x} \) or \( P(x) = {}_nC_x \cdot p^x \cdot (1 - p)^{n - x} \)
Where:
- \( P(x) \) is the probability of exactly \( x \) successes
- \( n \) is the number of trials
- \( x \) is the number of successes
- \( p \) is the probability of success on a single trial
- \( 1 - p \) is the probability of failure
- \( \frac{n!}{x!(n - x)!} \) is the number of combinations (n choose x)
- \( {}_nC_x \) also represents the number of combinations of \( n \) items taken \( x \) at a time
Mean for the Binomial Distribution
The mean of a binomial distribution represents the expected number of successes in a fixed number of independent trials, each with the same probability of success. It indicates the long-run average outcome if the experiment were repeated many times.
\( \mu = n \cdot p \)
Where:
- \( \mu \) is the mean (expected value)
- \( n \) is the number of trials
- \( p \) is the probability of success on a single trial
Variance for the Binomial Distribution
The variance of a binomial distribution measures how much the number of successes is expected to vary from the mean across repeated trials. It reflects the spread or dispersion of possible outcomes in a binomial experiment.
\( \sigma^2 = n \cdot p \cdot q \)
Where:
- \( \sigma^2 \) is the variance
- \( n \) is the number of trials
- \( p \) is the probability of success
- \( q \) is the probability of failure, where \( q = 1 - p \)
Standard Deviaiton for the Binomial Distribution
The standard deviation of a binomial distribution measures the average distance of the number of successes from the mean. It indicates how much the outcomes typically vary in a binomial experiment. It is the square root of the variance and gives a sense of the spread of the distribution.
\( \sigma = \sqrt{n \cdot p \cdot q} \)
Where:
- \( \sigma \) is the standard deviation
- \( n \) is the number of trials
- \( p \) is the probability of success
- \( q \) is the probability of failure, where \( q = 1 - p \)
Authors
"5.5: Chapter 5 Formulas" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY 4.0