Stat 321 – Lecture 12Last Time – Discrete RVsOther propertiesSlide 4Functions of random variablesPop QuizExchange with Neighbor“Solutions”Example 1Slide 10Slide 11Slide 12Slide 13For TuesdayStat 321 – Lecture 12Discrete Random Variables (cont.)Binomial Probability Distribution (2.4)Last Time – Discrete RVsRandom variable = maps every outcome in sample space to a numerical valueDiscrete = can “list” every possible valueProbability mass function (pmf)List every outcome, assign probabilityLine graph/probability histogramExpected value E(X) = xi P(X=xi )Long-run averageCumulative distribution function F(x) = P(X < x)P(X > x) = 1-F(x)P(a < X < b) = F(b) –F(a-)Other propertiesVarianceInterpretationShort-cut formula: V(X) = E(X2) – [E(X)]2 )()()(2xXPxXVxiE(Y) = 3(.15)+ 6(.35) + 9(.5) = 7.05V(Y)=(3-7.05)2(.15)+(6-7.05)2(.35)+(9-7.05)2(.5)= 4.75SD(Y) = 2.18E(X) = 1(.3)+3(.1)+4(.05)+6(.15)+12(.4) = 6.5E(X2)=1(.3)+9(.1)+16(.05)+36(.15)+144(.4)= 65V(X) = 65 – (6.52) = 22.75SD(X) = 4.17Functions of random variablesIn general, E(h(X)) = h(x)P(X=x)E.g., E(X2)But what about h(X) = aX + b?E.g., h(X) = 5/9X – 160/9Cut in half and subtract 18SLO temps, mean 72, std dev 6Cut in half and subtract 18? 22, -15??When have a linear functionE(aX+b) = aE(X)+b, SD(aX+b) = |a|SD(X)Pop QuizTake out a blank piece of paperIndividualMultiple choice, 3 options (A, B, C)10 questionsExchange with NeighborRecord number correct“Solutions”1) C2) A3) B4) C5) C6) C7) B8) A9) C10) CExample 1Probability DistributionX = number correct out of 4 multiple choice questions (3 options each)Possible values for X?Probability for each value?What assumptions are you making?Example 1x 0 1 2 3 4p(x) .20 .40 .30 .10 .01P(X>2) = .3+.1+.01=.41P(X>2) = 1-P(X<1)= 1-.2-.4=.4Example 1General formConditionsTwo possible outcomesConstant probability of success, pTrials are independentRandom variable counts the number of successes in a fixed number of trials, nExample 1What if there are 10 questions?How find P(passing)P(X > 5) = 1-P(X < 4)TableMinitabExample 1For TuesdayHW 4…Exam 1Mean = .82, Median = .84Solutions in BlackboardCourse Avg updatedCorrect answers vs. highest scoring answers/
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