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Unformatted text preview: . if X is in feet, then Var(X ) is in feet2 and
σX is again in feet.
As mentioned earlier, the variance of X is a measure of how spread out the pmf of X is. If a
constant b is added to X , the pmf of the resulting random variable X + b is obtained by shifting the
pmf of X by b. Adding the constant increases the mean by b, but it does not change the variance,
because variance measures spread around the mean:
E [X + b] = E [X ] + b
Var(X + b) = E [(X + b − E [X + b])2 ] = E [(X + b − E [X ] − b)2 ] = E [(X − E [X ])2 ] = Var(X ).
If X is multiplied by a constant a, the pmf of the resulting random variable is spread out by a
factor a. The mean is multiplied by a, and the variance is multiplied by a2 :
E [aX ] = aE [X ]
Var(aX ) = E [(aX − E [aX ])2 ] = E [(aX − aE [X ])2 ] = a2 E [(X − E [X ])2 ] = a2 Var(X ).
Combining the two observations just made yields that:
E [aX + b] = aE [X ] + b and Var(aX + b) = Var(aX ) = a2 Var(X ).
The random variable XσX X is called the standardized version of X . The standardized random
variable has mean zero and variance...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
- Spring '08
- The Land