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 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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 Binomial mean =
 Expected Value =
 Where:
 μ is the expected value.
 E(x) denotes Expected Value.
 Σ called Sigma is the sum or total.
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 x is each variable data value.
P(x) is the probability for each x.
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 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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 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
 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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 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
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1)
2)
3)
4)
5)
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
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 Calculate discrete variance for these numbers
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X Probability
0 .65
1 .10
2 .20
3 .05
X – mean square X*p(x)
0 - .65 -.65
2 .4225(.65)
1 - .65 .35
2 .1225(.10)
2 - .65 1.35
2 1.8225(.2)
3 - .65 2.35
2 5.5225(.05)
Total = .9275
 Mean = 0(.65) + 1(.10) + 2(.20) + 3(.05) = .65
 Variance is .9275
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
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Calculate discrete variance for these numbers
Variance = Sum(x 2 multiply Probability) – mean 2
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X Probability X 2 X 2 (Probability)
0 .65 0 0
1 .10 1 0.1
2 .20 4 0.8
3 .05 9 0.45
 Total = 1.35
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Mean = 0(.65) + 1(.10) + 2(.20) + 3(.05) = .65
 Excel: =SUMPRODUCT(a2:a5,b2:b5) = 0.65
Mean 2 = .4225
Variance is 1.35 - .4225 = 0.9275
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 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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