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Chapter 7: Q. 7.48 (page 363)

If Y=aX+bwhere a and b are constants, express the moment generating function of Y in terms of the moment generating function of X.

Short Answer

Expert verified

TheetbMX(at) moment generating function and in the fourth equality we used linearity of expectation.

Step by step solution

01

Given Information

We need to find the moment generating function of Y=aX+b.

02

Explanation 

We need to find the moment generating function of Y=aX+bwhere a,b鈭�鈩�.

We evaluate to get

MY(t)=EetY

Substitute,

=Eet(aX+b).

03

Explanation

Simplify,

=EeatXetb

=etbEeatX

Evaluate that,

=etbMX(at)

In first and last equality we used the definition of the moment generating function and in the fourth equality we used linearity of expectation.

04

Final answer

The etbMX(at)moment generating function and in the fourth equality we used linearity of expectation.

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Most popular questions from this chapter

There are n items in a box labeled H and m in a box labeled T. A coin that comes up heads with probability p and tails with probability 1 鈭� p is flipped. Each time it comes up heads, an item is removed from the H box, and each time it comes up tails, an item is removed from the T box. (If a box is empty and its outcome occurs, then no items are removed.) Find the expected number of coin flips needed for both boxes to become empty. Hint: Condition on the number of heads in the first n + m flips.

Consider the following dice game: A pair of dice is rolled. If the sum is7,then the game ends and you win 0.If the sum is not 7,then you have the option of either stopping the game and receiving an amount equal to that sum or starting over again. For each value ofi,i=2,...,12, find your expected return if you employ the strategy of stopping the first time that a value at least as large as i appears. What value ofileads to the largest expected return? Hint: Let Xidenote the return when you use the critical value i.To computeE[Xi], condition on the initial sum.

Show that Zis a standard normal random variable and if Yis defined by Y=a+bZ+cZ2, then

蚁(Y,Z)=bb2+2c2

Let Z be a standard normal random variable,and, for a 铿亁ed x, set

X={ZifZ>x0otherwise

Show thatE[X]=12蟺e鈭�x2/2.

7.2. Suppose that Xis a continuous random variable with

density function f. Show that E[IX-a鈭�]is minimized

when ais equal to the median of F.

Hint: Write

E[IX-al]=|x-a|f(x)dx

Now break up the integral into the regions where x<a

and where x>a, and differentiate.
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