# Moment generating function problem

From the pdf of X, f(x) = 1/8 e^-x/8, x > 0, find the mgf of Y=X/4 +1. What is then the value of P(2.3 < Y < 4.1)?

## Homework Equations

Moment generating function of exponential distribution

## The Attempt at a Solution

I have the mgf of X, which is 1/8 / (1/8 - t). I have also worked out the mgf of Y, which is e^t (1/8 (1/8 - t/4)), I think. The last part of this problem I've yet to resolve. Please do help!

Last edited:

Ray Vickson
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From the pdf of X, f(x) = 1/8 e^-x/8, x > 0, find the mgf of Y=X/4 +1. What is then the value of P(2.3 < Y < 4.1)?

## Homework Equations

Moment generating function of exponential distribution

## The Attempt at a Solution

I have the mgf of X, which is 1/8 / (1/8 - t). I have also worked out the mgf of Y, which is e^t (1/8 (1/8 - t/4)), I think. The last part of this problem I've yet to resolve. Please do help!

Please state exactly what definition of mgf you are using; the one that I use (standard, I think) gives a very different result from yours.

As to the second question: I don't think the mgf has any relevance here; you need to relate the interval probabilities of Y to those of X, and use the density of X to compute the result. In other words, {a <= Y <= b} is the same as {a1 <= X <= b1} for some a1 and b1 related to a and b, and you know how to calculate P{a1 <= X <= b1}.

RGV

I'm using the mgf of exponential: lambda/(lambda -t). So, to obtain the probability, simply integrate the probability density function of X with the values of a1 and b1, is that it?