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Unformatted text preview: j ) = ij. Each 26 CHAPTER 2. DISCRETE-TYPE RANDOM VARIABLES 4/36 p (k)
5 10 20 15 25 30 35 Figure 2.3: The pmf for the product of two fair dice
of the 36 possible values of (X1 , X2 ) has probability 1/36, so we have
E [Y ] = = = 1
36 6 6 ij
i=1 j =1 1
36 6 i
36 6 = j j =1 7
2 2 = 49
4 Variance and standard deviation Suppose you are to be given a payment, with the size of
the payment, in some unit of money, given by either X or by Y, described as follows. The random
variable X is equal to 100 with probability one, whereas pY (100000) = 1000 and pY (0) = 1000 .
Would you be equally happy with either payment? Both X and Y have mean 100. This example
illustrates that two random variables with quite diﬀerent pmfs can have the same mean. The pmf
for X is concentrated on the mean value, while the pmf for Y is considerably spread out.
The variance of a random variable X is a measure of how spread out the pmf of X is. Letting
µX = E [X ], the variance is deﬁned by:
Var(X ) = E [(X − µX )2...
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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