Discrete Probability Distributions GrowingKnowing.com © 2011 GrowingKnowing.com © 2011

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 © 2011

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 © 2011

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 © 2011

Expected value example What is the expected value for the discrete random distribution where variable x has these values: x     P(x) 0     .50 1     .30 2     .10 3     .10 Answer = 0(.5) + 1(.3) + 2(.1) + 3(.1) = .8 EXCEL: =SUMPRODUCT(A1:A4,B1:B4) 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 © 2011

Variance in Discrete Probability Distributions Binomial variance = σ2 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 x (1 - .3) = 30 x (.7) = 21 GrowingKnowing.com © 2011

Discrete Variance Calculate the mean (i.e. expected value) Subtract the mean from each value of X Square result Multiply by the probability for that value of X Total the result for the variance GrowingKnowing.com © 2011

Discrete Variance – Hard way Calculate discrete variance for these numbers X    Probability 0 .65 1 .10 2 .20 3 .05 Total = .9275 Mean = 0(.65) + 1(.10) + 2(.20) + 3(.05) = .65 Variance is .9275 X – mean (X-mean)2 (X-mean)2*p(x) 0 - .65 .4225 .4225(.65) 1 - .65 .1225 .1225(.10) 2 - .65 1.8225 1.8225(.2) 3 - .65 5.5225 5.5225(.05) GrowingKnowing.com © 2011

Discrete Variance – Easy way Calculate discrete variance for these numbers Variance = Sum(x2 multiply Probability) – mean2 X    Probability X2 X2(Probability) 0 .65 0 0 1 .10 1 0.1 2 .20 4 0.8 3 .05 9 0.45 Total = 1.35 Mean = 0(.65) + 1(.10) + 2(.20) + 3(.05) = .65 Excel: =SUMPRODUCT(a2:a5,b2:b5) = 0.65 Mean2 = .4225 Variance is 1.35 - .4225 = 0.9275 GrowingKnowing.com © 2011

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 © 2011