Mean, Variance, Standard Deviation, and Expectation by variablepitch347

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									Mean, Variance, Standard Deviation, and Expectation
Carla J. Adamson

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Objective
Find the mean, variance, standard deviation, and expected value for a discrete random variable.

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Background
Mean, variance, and standard deviation for a probability distribution are not calculated the same way as samples or populations are calculated. Rounding Rule for mean, variance, and standard deviation: round to one more decimal place than the outcome (random variable). When fractions are used, they should be reduced to lowest terms.
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Mean
In chapter 3, the mean for a sample or population was computed by adding the values and dividing by N. Since in a probability distribution, there is no N, then we cannot use that formula. Multiply the random variable times its respective probability, and then add all the products
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Example page 252
Outcome X Probability P(X) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6

Find the mean of the number of spots that appear when a die is tossed. μ = Σ(X*P(X)) = 1*1/6+2*1/6+3*1/6+4*1/6+5*1/6+6*1/6 = 21/6 =3.5
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Variance and Standard Deviation
For the same reason as mean, we need a new way to calculate variance for a probability distribution. Variance: Square the random variables, multiply by the respective probabilities, add the products, and then subtract the square of the mean (shortcut). Standard deviation is the square root of the variance.
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Example page 254
Outcome X

1 1 6

2 1 6

3 1 6

4 1 6

5 1 6

6 1 6

Probability P(X)

σ2=Σ[X2*P(X)] – μ2 = 1*1/6+4*1/6+9*1/6+16*1/6+25*1/6+36*1/6 – 3.52 = 91/6 – 3.52 = 2.9 σ = square root(2.9) = 1.7
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Heads up
Do not use a rounded mean in variance calculation. Do not use a rounded variance in standard deviation calculation.

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Expected Value
Expected Value of a discrete random variable of a probability distribution is the mean of the variable. Generally, expected value comes in when there is money on the line.

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