DefinitionsExpected ValueTheexpected valueof a discrete random variableX(denotedE(X), μ)with PMFf(x) is the value that you would expect in the long-run throughrepeating your random experiment an infinite number of times. Theexpected value ofXis given by:E(X) =Xx∈Xx·f(x)Revew of Summation Rules:∑ni=1c=n·cfor a constant c∑ni=1c·xi=c·(∑ni=1xi)∑ni=1i=n·(n+1)2September 23, 201738 / 56
Expected ValueExpected Value of a FunctionIfXis a discrete random variable with probability mass functionf(x),then for anyg(X) that is a real-valued function of X, then theexpectedvalue ofg(X)is given by:E[g(X)] =Xx∈Xg(x)·f(x)September 23, 201739 / 56
Variance ExampleExample 10A metal fabricating plant currently has five major piecesunder contract, each with a deadline for completion. LetXbe the numberof pieces completed by their deadlines. Suppose thatXis a randomvariable with probability mass function:X012345P(X=x)0.050.100.150.250.350.10Find the variance of the number of contracts they will be able to completeby their deadline.September 23, 201741 / 56
Variance ExampleExample 10Interpretation: On average, the company will completesquaredcontracts more or less than their average ofby the deadline.September 23, 201742 / 56
Variance ExampleExample 12The standard deviation for example 12 is:σ=√1.7475 = 1.32That is, on “average” (typically), the company will be able to completewithin 1.32 contracts of the mean on time.September 23, 201744 / 56
Expected Value ExamplesExample 13There is a bowl containing 30 cashews, 20 pecans, 25almonds, and 25 walnuts. Cal randomly picks 3 nuts to eat. What is theexpected number of cashews Cal will eat? The variance?