Chapter 3 Sections 4 to 7.pdf - 3.4 The Expected Value of a...

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3.4 The Expected Value of a Function of a Random Variable Often times in the real world, we wish to look at a function of our variables or transform them. Changing feet into meters, Fahrenheit into centigrade, etc. In this section we seek to determine the expected value of this new variable. If we transform a random variable X by some function ( ) Y g X , it should be clear that Y is a random variable. Suppose that we have a discrete random variable X with pmf given below. We now let Y X 2 5 . What is the expected value of Y ? By definition [ ] ( ) Y E Y yf y . So to determine the [ ] E Y , we will find ( ) Y f y and then compute [ ] ( ) Y E Y yf y . So, Y E Y yf y 1 2 54 [ ] ( ) (1 3 7 9 11 13) 5 8 8 8 . We also not that, Y X X x x Support X E Y yf y x f x g x f x 4 2 ( ) [ ] ( ) (2 5) ( ) ( ) ( )  Theorem: Given a discrete random variable X and the linear transformation Y g X aX b ( ) , X x Support X E Y E g X g x f x ( ) [ ] [ ( )] ( ) ( ) . Proof: Since Y g X aX b ( ) is a 1 to 1 function, this will be rather straight forward. Y X E Y yf y ax b P Y ax b ax b P X x g x f x [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) Note that the argument above would work with any function ( ) Y g X that is 1 to 1. Theorem: Given a discrete random variable X and the linear transformation Y g X aX b ( ) , E Y E aX b aE X b [ ] [ ] [ ] . Proof: [ ] ( ) ( ) ( ) ( ) ( ) ( ) E aX b ax b f x ax f x b f x ( ) ( ) [ ] X a xf x b f x aE X b a b Thus, the expected value of a random variable is a linear operator . x ( ) X f x y ( ) Y f y -2 1/8 1 1/8 -1 1/8 Y X 2 5 3 1/8 0 2/8 5 2/8 1 1/8 7 1/8 2 1/8 9 1/8 3 1/8 11 1/8 4 1/8 13 1/8
Example: If a random variable X has mean 4 , determine the mean of Y , where 2 3 Y X 2 3 11 Y X or [ ] [2 3] 2 [ ] 3 11 E Y E X E X If we think about this last example, it seems like a no brainer that it is true. Suppose that the average score is 4. I now double everybody’s score. The new average is 8. I now add 3 to everybody’s already doubled score. The new average is 11. In this linear case, notice if we consider ( ) 2 3 Y g X X , then [ ] [ ( )] 2 3 ( [ ]) E Y E g X X g E X . We would naturally wonder if that statement always holds. That is, will it always be true that If ( ) Y g X then [ ] ( [ ]) E Y g E X . The answer is no, this is not always true. How will we show this? Suppose that we have a discrete random variable X with pmf given below. The expected value of X can be determined by xf x ( ) 7 / 8 . We now let 2 Y X . What is the expected value of Y ? By definition [ ] ( ) Y E Y yf y . So to determine the [ ] E Y , we will find ( ) Y f y and then compute [ ] ( ) Y E Y yf y . [ ] 7 / 8 E X [ ] 35/ 8 E Y We note that E X E Y
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