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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

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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.

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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.

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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.

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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

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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

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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

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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

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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

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