# 4.2 (cont.) Standard Deviation of a Discrete Random Variable

View again

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Information Report
Category:

## Documents

Published:

Views: 24 | Pages: 31

Share
Description
4.2 (cont.) Standard Deviation of a Discrete Random Variable. First center (expected value) Now - spread. 4.2 (cont.) Standard Deviation of a Discrete Random Variable. Measures how “spread out” the random variable is. Data Histogram measure of the center: sample mean x measure of spread:
Transcript
4.2 (cont.) Standard Deviation of a Discrete Random VariableFirst center (expected value)Now - spread4.2 (cont.) Standard Deviation of a Discrete Random VariableMeasures how “spread out” the random variable isDataHistogrammeasure of the center: sample mean xmeasure of spread: sample standard deviation sRandom variableProbability Histogrammeasure of the center: population mean mmeasure of spread: population standard deviation sSummarizing data and probabilityExample
• x 0 100
• p(x) 1/2 1/2 E(x) = 0(1/2) + 100(1/2) = 50
• y 49 51
• p(y) 1/2 1/2E(y) = 49(1/2) + 51(1/2) = 50VariationThe deviations of the outcomes from the mean of the probability distribution xi - µ2 (sigma squared) is the variance of the probability distributionVariationVariance of discrete random variable XEconomicScenarioProfit(\$ Millions)ProbabilityXPGreat100.20x1P(X=x1)5Good0.40x2P(X=x2)OK10.25x3P(X=x3)Lousy-40.15x4P(X=x4)VariationExample2 = (x1-µ)2 · P(X=x1) + (x2-µ)2 · P(X=x2) + (x3-µ)2 · P(X=x3) + (x4-µ)2 · P(X=x4) = (10-3.65)2 · 0.20 + (5-3.65)2 · 0.40 + (1-3.65)2 · 0.25 + (-4-3.65)2 · 0.15 = 19.32753.653.653.653.65P. 207, Handout 4.1, P. 4Standard Deviation: of More Interest then the Variance2 = 19.3275Standard Deviation, or SD, is the standard deviation of the probability distributionProbability Histogram = 4.40µ=3.65Finance and Investment Interpretation
• X = return on an investment (stock, portfolio, etc.)
• E(x) = m = expected return on this investment
• sis a measure of the risk of the investment
• ExampleA basketball player shoots 3 free throws. P(make) =P(miss)=0.5. Let X = number of free throws made.Expected Value of a Random VariableExample: The probability model for a particular life insurance policy is shown. Find the expected annual payout on a policy.We expect that the insurance company will pay out \$200 per policy per year.13© 2010 Pearson Education Standard Deviation of a Random VariableExample: The probability model for a particular life insurance policy is shown. Find the standard deviation of the annual payout.14© 2010 Pearson Education 68-95-99.7 Rule for Random VariablesFor random variables x whose probability histograms are approximately mound-shaped:
• P(m - s  x  m + s)  .68
• P(m - 2s  x  m + 2s)  .95
• P(m -3s  x  m + 3s)  .997
• (m - 1s, m + 1s) (50-5, 50+5) (45, 55)P(m - s  X  m + s) = P(45  X  55)=.048+.057+.066+.073+.078+.08+.078+.073+ .066+.057+.048=.724Rules for E(X), Var(X) and SD(X):adding a constant a
• If X is a rv and a is a constant:
• E(X+a) = E(X)+a
• Example: a = -1
• E(X+a)=E(X-1)=E(X)-1
• Rules for E(X), Var(X) and SD(X): adding constant a (cont.)
• Var(X+a) = Var(X)
• SD(X+a) = SD(X)
• Example: a = -1
• Var(X+a)=Var(X-1)=Var(X)
• SD(X+a)=SD(X-1)=SD(X)
• EconomicScenarioProfit(\$ Millions)XProbabilityEconomicScenarioProfit(\$ Millions)X+2ProbabilityPPGreat100.20Great10+20.20x1x1+2P(X=x1)P(X=x1)55+2Good0.40Good0.40x2x2+2P(X=x2)P(X=x2)OK10.25OK1+20.25x3x3+2P(X=x3)P(X=x3)Lousy-40.15Lousy-4+20.15x4x4+2P(X=x4)P(X=x4)E(x + a) = E(x) + a; SD(x + a)=SD(x); let a = 2 = 4.40 = 4.40m=5.65m=3.65New Expected ValueLong (UNC-CH) way: (compute from “scratch”)E(x+2)=12(.20)+7(.40)+3(.25)+(-2)(.15) = 5.65Smart (NCSU) way:a=2; E(x+2) =E(x) + 2 = 3.65 + 2 = 5.65New Variance and SDLong (UNC-CH) way: (compute from “scratch”)Var(X+2)=(12-5.65)2(0.20)+… +(-2+5.65)2(0.15) = 19.3275SD(X+2) = √19.3275 = 4.40Smart (NCSU) way:Var(X+2) = Var(X) = 19.3275SD(X+2) = SD(X) = 4.40Rules for E(X), Var(X) and SD(X): multiplying by constant b
• E(bX)=b E(X)
• Var(b X) = b2Var(X)
• SD(bX)= |b|SD(X)
• Example: b =-1
• E(bX)=E(-X)=-E(X)
• Var(bX)=Var(-1X)=
• =(-1)2Var(X)=Var(X)
• SD(bX)=SD(-1X)=
• =|-1|SD(X)=SD(X)Expected Value and SD of Linear Transformation a + bxLet X=number of repairs a new computer needs each year. Suppose E(X)= 0.20 and SD(X)=0.55The service contract for the computer offers unlimited repairs for \$100 per year plus a \$25 service charge for each repair.What are the mean and standard deviation of the yearly cost of the service contract?Cost = \$100 + \$25XE(cost) = E(\$100+\$25X)=\$100+\$25E(X)=\$100+\$25*0.20== \$100+\$5=\$105SD(cost)=SD(\$100+\$25X)=SD(\$25X)=\$25*SD(X)=\$25*0.55==\$13.75Addition and Subtraction Rules for Random Variables
• E(X+Y) = E(X) + E(Y);
• E(X-Y) = E(X) - E(Y)
• When X and Y are independent random variables:
• Var(X+Y)=Var(X)+Var(Y)
• SD(X+Y)=
• SD’s do not add:SD(X+Y)≠ SD(X)+SD(Y)
• Var(X−Y)=Var(X)+Var(Y)
• SD(X −Y)=
• SD’s do not subtract:SD(X−Y)≠ SD(X)−SD(Y)SD(X−Y)≠ SD(X)+SD(Y)Motivation forVar(X-Y)=Var(X)+Var(Y)
• Let X=amount automatic dispensing machine puts into your 16 oz drink (say at McD’s)
• A thirsty, broke friend shows up.
• Let Y=amount you pour into friend’s 8 oz cup
• Let Z = amount left in your cup; Z = ?
• Z = X-Y
• Var(Z) = Var(X-Y) =
• Var(X)Has 2 components+ Var(Y)Example: rv’s NOT independent
• X=number of hours a randomly selected student from our class slept between noon yesterday and noon today.
• Y=number of hours the same randomly selected student from our class was awake between noon yesterday and noon today. Y = 24 – X.
• What are the expected value and variance of the total hours that a student is asleep and awake between noon yesterday and noon today?
• Total hours that a student is asleep and awake between noon yesterday and noon today = X+Y
• E(X+Y) = E(X+24-X) = E(24) = 24
• Var(X+Y) = Var(X+24-X) = Var(24) = 0.
• We don't add Var(X) and Var(Y) since X and Y are not independent.
• Pythagorean Theorem of Statistics for Independent X and Ya2+b2=c2Var(X+Y)=Var(X+Y)c2+Var(Y)Var(X)Var(X)ca2aSD(X+Y)SD(X)a + b ≠ cSD(X)+SD(Y) ≠SD(X+Y)bSD(Y)b2Var(Y)Pythagorean Theorem of Statistics for Independent X and Y32 + 42 = 52Var(X)+Var(Y)=Var(X+Y)25=9+16Var(X)Var(X+Y)593SD(X+Y)SD(X)3 + 4 ≠ 5SD(X)+SD(Y) ≠SD(X+Y)4SD(Y)16Var(Y)Example: meal plans
• Regular plan: X = daily amount spent
• E(X) = \$13.50, SD(X) = \$7
• Expected value and stan. dev. of total spent in 2 consecutive days?
• E(X1+X2)=E(X1)+E(X2)=\$13.50+\$13.50=\$27
• SD(X1 + X2) ≠ SD(X1)+SD(X2) = \$7+\$7=\$14Example: meal plans (cont.)
• Jumbo plan for football players Y=daily amount spent
• E(Y) = \$24.75, SD(Y) = \$9.50
• Amount by which football player’s spending exceeds regular student spending is Y-X
• E(Y-X)=E(Y)–E(X)=\$24.75-\$13.50=\$11.25
• SD(Y ̶ X) ≠ SD(Y) ̶ SD(X) = \$9.50 ̶ \$7=\$2.50For random variables, X+X≠2X
• Let X be the annual payout on a life insurance policy. From mortality tables E(X)=\$200 and SD(X)=\$3,867.
• If the payout amounts are doubled, what are the new expected value and standard deviation?
• Double payout is 2X. E(2X)=2E(X)=2*\$200=\$400
• SD(2X)=2SD(X)=2*\$3,867=\$7,734
• Suppose insurance policies are sold to 2 people. The annual payouts are X1 and X2. Assume the 2 people behave independently. What are the expected value and standard deviation of the total payout?
• E(X1 + X2)=E(X1) + E(X2) = \$200 + \$200 = \$400
• The risk to the insurance co. when doubling the payout (2X) is not the same as the risk when selling policies to 2 people.
Recommended

35 pages

7 pages

4 pages

15 pages

10 pages

10 pages

10 pages

10 pages

5 pages

1 page

8 pages

8 pages

7 pages

1 page

### Biocatalytic synthesis of ethyl ( R)-2-hydroxy-4-phenylbutyrate with Candida krusei SW2026: A practical process for high enantiopurity and product titer

6 pages

View more...

#### Developmental Psychology

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us. Thanks to everyone for your continued support.

No, Thanks
SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...

Sign Now!

We are very appreciated for your Prompt Action!

x