# Given marginal pdfs of X and Y, find pdf of Z=X-Y

## Homework Statement

The probability density function of the random variables X and Y are given by:

$$f_1(x)= \begin{cases} 2 & -\frac{1}{4}\le x\le \frac{1}{4} \\ 0 & \text{elsewhere} \end{cases}$$
and
$$f_2(y) \begin{cases} \frac{1}{2} & 0\le y \le 2 \\ 0 & \text{elsewhere} \end{cases}$$

respectively.

a) Find the probability density function of the random variable Z=X-Y .
b) What is the probability that Z will assume a value greater than zero?

Not sure yet.

## The Attempt at a Solution

There isn't an example like this in my book. I'm not sure how to go from marginals to the new variable thing, which I couldn't solve in an ordinary manner anyway! Sad sad sad. Am I supposed to make the marginals into a regular f(x,y), or is there some direct way to get to the Z?

## Answers and Replies

D H
Staff Emeritus
Science Advisor
I assume your book tells you how to compute the distribution of a sum of random variables such as W=X+Y.

One way to look at this is to invent a new random variable U=-Y. (Use Z=X-Y=X+(-Y)=X+U.) What does the distribution of this variable U look like? of X+U?

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed

## Homework Statement

The probability density function of the random variables X and Y are given by:

$$f_1(x)= \begin{cases} 2 & -\frac{1}{4}\le x\le \frac{1}{4} \\ 0 & \text{elsewhere} \end{cases}$$
and
$$f_2(y) \begin{cases} \frac{1}{2} & 0\le y \le 2 \\ 0 & \text{elsewhere} \end{cases}$$

respectively.

a) Find the probability density function of the random variable Z=X-Y .
b) What is the probability that Z will assume a value greater than zero?

Not sure yet.

## The Attempt at a Solution

There isn't an example like this in my book. I'm not sure how to go from marginals to the new variable thing, which I couldn't solve in an ordinary manner anyway! Sad sad sad. Am I supposed to make the marginals into a regular f(x,y), or is there some direct way to get to the Z?

Unless you are given more information you cannot do the question:you need to know something about the joint distribution of the pair (X,Y). In particular, are X and Y independent? If they *are* independent, just let Y1 = -Y and look at X+Y1. The distribution of Y1 is easy to get, and surely the distribution of X+Y1 must be obtainable from material in your textbook or notes.

RGV

Unless you are given more information you cannot do the question:you need to know something about the joint distribution of the pair (X,Y). In particular, are X and Y independent? If they *are* independent, just let Y1 = -Y and look at X+Y1. The distribution of Y1 is easy to get, and surely the distribution of X+Y1 must be obtainable from material in your textbook or notes.

RGV

What I typed is all I have.

D H
Staff Emeritus
Science Advisor
So assume they are independent. As both Ray and I noted, your text or notes must have something to say about the sum of two independent random variables.

Hmm, it looks like if they are independent then $f(x,y)=f_1(x)f_2(y)$
From there, it's like any other random variable problem. Thanks for the suggestion. :)

I like Serena
Homework Helper
Hi Arcana! For adding or subtracting independent distributions, we have the convolution rule for distributions.

Suppose X and Y are independent probability distributions with probability density functions fX(x) and fY(y), and cumulative probability function FX(x) and FY(y).

If U=X+Y, then
$$P(U \le u) = P(X + Y \le u) = \int_{-\infty}^{\infty} f_X(x) P(x+Y \le u) \textrm{ d}x = \int_{-\infty}^{\infty} f_X(x) P(Y \le u - x) \textrm{ d}x$$
so
$$P(U \le u) = \int_{-\infty}^{\infty} f_X(x) F_Y(u-x) \textrm{ d}x$$

And if you want to know the probability density of U, we have:
$$f_U(u)= {d \over du}F_U(u) = {d \over du}P(U \le u)$$

great, thanks!