Post on 02-Jan-2016
GrowingKnowing.com © 2011
1GrowingKnowing.com © 2011
Expected valueExpected value is a weighted meanExample
You put your data in categories by productYou build a frequency and relative frequency chartYou see Product A has a relative frequency of .5You 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 = 50Formula is Expected Value = n x p
GrowingKnowing.com © 2011 2
Formula Expected ValueBinomial 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 3
ExamplesWhat 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 valueWe do not need functions for multiplication or
addition.
GrowingKnowing.com © 2011 4
Expected value exampleWhat 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
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 5
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(1 - .3) = 30(.7) =
21
GrowingKnowing.com © 2011 6
Discrete Variance
1)Calculate the mean (i.e. expected value)2)Subtract the mean from each value of X3)Square result4)Multiply by the probability for that value of X5)Total the result for the variance
GrowingKnowing.com © 2011 7
Discrete Variance – Hard wayCalculate 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) = .65Variance is .9275
GrowingKnowing.com © 2011 8
X – mean square X*p(x)0 - .65 -.652 .4225(.65) 1 - .65 .352 .1225(.10)2 - .65 1.352 1.8225(.2)3 - .65 2.352
5.5225(.05)
Discrete Variance – Easy wayCalculate discrete variance for these numbersVariance = 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 = .4225Variance is 1.35 - .4225 = 0.9275
GrowingKnowing.com © 2011 9
Discrete Standard DeviationTake 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=.3Step 1: calculate the variance using formula np(1-p)
=200*.3*(1-.3) = 60(.7) = 42Step 2: take square root of variance.
=sqrt(42) = 6.48
GrowingKnowing.com © 2011 10