The variance of a discrete random variable, denoted by V (X), is defined to be
That is, V (X) is the average squared distance between X and its mean. Variance is a measure of dispersion, telling us how “spread out” a distribution is. For our simple random variable, the variance is
V (X) = (1− 3.25)2 (.25) + (2 − 3.25)2 (.25) + (5 − 3.25)2 (.50) = 3.1875.
A slightly easier way to calculate the variance is to use the well-known identity
V (X) = E(X2) − (E(X) )2.
Visually, this method requires a table with three columns: x, f(x), and x2.
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x
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f(x)
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x2
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1
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.25
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12 = 1
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2
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.25
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22 = 4
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5
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.50
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52 = 25
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First we calculate
E(X) = 1(.25) + 2(.25) + 5(.50) = 3.25 and
E(X2) = 1(.25) + 4(.25) + 25(.50) = 13.75. Then
V (X) = 13.75 − (3.25)2 = 3.1875.
It can be shown that if a and b are constants, then
V (a + bX) = b2V (X).
In other words, adding a constant a to a random variable does not change its variance, and multiplying a random variable by a constant b causes the variance to be multiplied by b2.
Another common measure of dispersion is the standard deviation, which is merely the positive square root of the variance,
Mean and Variance of a Sum of Random Variables
Expectation is always additive; that is, if X and Y are any random variables, then
E(X + Y ) = E(X) + E( Y ).
If X and Y are independent random variables, then their variances will also add:
V (X + Y) = V (X) + V ( Y ) if X, Y independent.
More generally, if X and Y are any random variables, then
V (X + Y) = V (X) + V ( Y ) + 2Cov(X, Y )
where Cov(X, Y ) is the covariance between X and Y,
Cov(X, Y ) = E( (X− E(X)) ( Y − E( Y )) ).
If X and Y are independent (or merely uncorrelated) then Cov(X, Y ) = 0. This additive rule for variances extends to three or more random variables; e.g.,
V (X + Y + Z) = V (X) + V ( Y ) + V (Z) +2Cov(X, Y ) + 2Cov(X, Z) + 2Cov(Y, Z)
with all covariances equal to zero if X, Y , and Z are mutually uncorrelated.