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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
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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.
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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.
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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.
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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 x (1 - .3) = 30 x (.7) = 21
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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
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Discrete Variance – Hard way
 Calculate discrete variance for these numbers
X – mean (X-mean)2 (X-mean)2*p(x)
 X Probability
0 - .65
.4225
.4225(.65)
 0 .65
1 - .65
.1225
.1225(.10)
 1
.10
2 - .65
1.8225
1.8225(.2)
 2
.20
3 - .65
5.5225
5.5225(.05)
 3
.05
Total =
.9275
 Mean = 0(.65) + 1(.10) + 2(.20) + 3(.05) = .65
 Variance is .9275
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Discrete Variance – Easy way
 Calculate discrete variance for these numbers
 Variance = Sum(x2 multiply Probability) – mean2
X Probability
0 .65
1 .10
2 .20
3 .05
Total =
X2
0
1
4
9
X2(Probability)
0
0.1
0.8
0.45
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
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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
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