1. Discrete Probability Distributions
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2. 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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3. 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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4. 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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5. Expected value example
What is the expected value for the discrete random distribution where variable x has these values:
x     P(x) 0                 .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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6. 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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7. 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
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8. Discrete Variance – Hard way
Calculate discrete variance for these numbers
X    Probability
Total =
Mean = 0(.65) + 1(.10) + 2(.20) + 3(.05) = .65
Variance is .9275
X – mean (X-mean) (X-mean)2*p(x)
(.65)
(.10)
(.2)
(.05)
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9. Discrete Variance – Easy way
Calculate discrete variance for these numbers
Variance = Sum(x2 multiply Probability) – mean2
X    Probability X X2(Probability)
Total =
Mean = 0(.65) + 1(.10) + 2(.20) + 3(.05) = .65
Excel: =SUMPRODUCT(a2:a5,b2:b5) = 0.65
Mean2 = .4225
Variance is =
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10. 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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