9. Random Variables, PDFs, Expected Value, Variance, and Standard Deviation
- Page ID
- 24958
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Contents 9A:
- Intro to random variables (https://youtu.be/E2NjGfMQSKc)
- RVs, PDFs, and EVs, definitions and Example 1 (https://youtu.be/Ryp52y5k6kk)
- RVs, PDFs, and EVs, Example 2 (https://youtu.be/ZVK-zVV6oXw)
- RVs, PDFs, and EVs, Example 3 (with combinations) (https://youtu.be/VHUBX01ZqhE)
- RVs, PDFs, and EVs, Example 4 (with a Venn diagram) (https://youtu.be/dUqcf_qj9MY)
- RVs, PDFs, and EVs, Example 5 (with a Bernoulli process) (https://youtu.be/NoJMYjg6VwI)
Prework 9A:
-
At each meeting of a club, one person is selected to draw a “lucky number.” That person gets the amount in dollars of the number drawn. The box contains 30 cards with the number 1, 14 cards with the number 4, three cards with the number 10, two cards with the number 20, and one card with the number 50. Let the random variable X be the number on the cards. Determine the PDF of X.
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An experimenter randomly selects two people from a group of 5 men and 4 women. A random variable X is the number of women selected. Find the probability density function of X and the expected value of X.
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The probability that a person owns an Iphone is 55%. Assume that 200 randomly chosen people are polled. What is the expected number of people who own an Iphone?
Solutions
- We make the PDF below.
Outcome X Probability draw a card with a 1 1 \(\frac{30}{50}\) draw a card with a 4 4 \(\frac{14}{50}\) draw a card with a 10 10 \(\frac{3}{50}\) draw a card with a 20 20 \(\frac{2}{50}\) draw a card with a 50 50 \(\frac{1}{50}\) -
The PDF is given in the first three columns below, with an extra column so we can compute the expected value.
Outcome X Probability Product 0 women, 2 men 0 \(\frac{C(4,0)C(5,2)}{C(9,2)}=\frac{10}{36}\) 0 1 woman, 1 man 1 \(\frac{C(4,1)C(5,1)}{C(9,2)}=\frac{20}{36}\) \(\frac{20}{36}\) 2 women, 0 men 2 \(\frac{C(4,2)C(5,0)}{C(9,2)}=\frac{6}{36}\) \(\frac{12}{36}\) We add up the entries in the final column to get that \(E[X]=\frac{32}{36}=\frac{8}{9}\).
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Let \(X=\) the number of people polled who do have an Iphone. Then \(X\) is the number of successes in a Bernoulli process, so \(E[X]=np=200\cdot .55=110\).
Contents 9B:
- Introduction to variance and standard deviation (https://youtu.be/WInD3N0hxio)
- Variance and standard deviation, example 1 (https://youtu.be/P8o4PrPr8kE)
- Variance and standard deviation, example 2, involves completing a pdf first (https://youtu.be/fqKoJux2G38)
- Variance and standard deviation, example 3 (part 1), involves combinations (https://youtu.be/4nxuHbt0Cq8)
- Variance and standard deviation, example 3 (part 2), involves combinations (https://youtu.be/TXAjvRzJy6s)
- Variance and standard deviation, example 4, shortcut for Bernoulli processes (https://youtu.be/-9eFPkqJy2E)
Prework 9B:
- Consider the random variable \(X\) and its pdf given in the table below. Determine the expected value, variance, and standard deviation of \(X\).
\(X\) Probability 10 .1 20 .5 30 .4 -
Consider the random variable \(Y\) and its pdf given in the table below. Determine \(E[Y]\), \(Var[Y]\), and \(\sigma\) for \(Y\).
\(Y\) Probability 8 \(\frac{1}{2}\) 12 \(\frac{1}{3}\) 24 \(\frac{1}{6}\) -
A softball player gets a hit 30% of the time, and fails to get a hit 70% of the time. Suppose she goes up to bat 1000 times in her career. What is the expected number of hits she gets? What is the variance and standard deviation of the number of hits she gets?
Solutions
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\(X\) Probability Product \((X-E[X])^2\) Product 10 .1 1 169 16.9 20 .5 10 9 4.5 30 .4 12 49 19.6 \(E[X]=23\) \(Var(X)=41, \sigma=\sqrt{41}\) -
\(Y\) Probability Product \((Y-E[Y])^2\) Product 8 \(\frac{1}{2}\) 4 16 8 12 \(\frac{1}{3}\) 4 0 0 24 \(\frac{1}{6}\) 4 144 24 \(E[Y]=12\) \(Var(Y)=32, \sigma=\sqrt{32}\) -
If we let \(X=\)the number of hits she gets in 1000 attempts, then we are in a situation in which we can use the Bernoulli shortcuts. Therefore \(E[X]=np=1000\cdot .3=300, Var(X)=np(1-p)=1000\cdot .3\cdot .7=210,\) and \(\sigma=\sqrt{210}\).