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Expected value Expected value is a weighted mean Example You put your data in categories by product You build a frequency and relative frequency chart You see Product A has a relative frequency of.5 You can now predict Product A sales! If clients buy 100 products a day, then Product A expected value for tomorrow’s sales is 100 x.5 = 50 Formula is Expected Value = n x p GrowingKnowing.com © 20112

Formula Expected Value Binomial mean = Expected Value = Where: μ is the expected value. E(x) denotes Expected Value. Σ called Sigma is the sum or total. x is each variable data value. P(x) is the probability for each x. GrowingKnowing.com © 20113

Examples What is the binomial mean if sample size is 100 and probability is.3? Mean = n * probability = 100 *.3 = 30 There are no Excel functions for expected value We do not need functions for multiplication or addition. GrowingKnowing.com © 20114

Expected value example What is the expected value for the discrete random distribution where variable x has these values: x P(x) Answer = 0(.5) + 1(.3) + 2(.1) + 3(.1) =.8 TIP: a common error is dividing by a count as you do for the arithmetic mean. There is NO division in expected value. GrowingKnowing.com © 20115

Variance in Discrete Probability Distributions Binomial variance = σ 2 is the variance n is the count for the size of the sample. p is the probability for the binomial. What is the binomial variance if n = 100 and probability is.3? Variance = np(1-p) = 100 x.3(1 -.3) = 30(.7) = 21 GrowingKnowing.com © 20116

Discrete Variance 1) Calculate the mean (i.e. expected value) 2) Subtract the mean from each value of X 3) Square result 4) Multiply by the probability for that value of X 5) Total the result for the variance GrowingKnowing.com © 20117

Discrete Variance – Hard way Calculate discrete variance for these numbers X Probability Total =.9275 Mean = 0(.65) + 1(.10) + 2(.20) + 3(.05) =.65 Variance is.9275 GrowingKnowing.com © X – mean square X*p(x) (.65) (.10) (.2) (.05)

Discrete Variance – Easy way Calculate discrete variance for these numbers Variance = Sum(x 2 multiply Probability) – mean 2 X Probability X 2 X 2 (Probability) Total = 1.35 Mean = 0(.65) + 1(.10) + 2(.20) + 3(.05) =.65 Excel: =SUMPRODUCT(a2:a5,b2:b5) = 0.65 Mean 2 =.4225 Variance is = GrowingKnowing.com © 20119

Discrete Standard Deviation Take the square root of the variance What is the standard deviation if the variance is 9 ? S.D. =SQRT(9) = 3 What is the binomial S.D. if n =200 and probability=.3 Step 1: calculate the variance using formula np(1-p) =200*.3*(1-.3) = 60(.7) = 42 Step 2: take square root of variance. =sqrt(42) = 6.48 GrowingKnowing.com ©