# Probability Mass Function of { Z | Z < 1 }, Z given Z is less than 1

Probability Mass Function of { Z | Z < 1 }, "Z given Z is less than 1"

## Homework Statement

Given Z = X + Y.

Find the probability density function of Z|Z < 1.

N.A.

## The Attempt at a Solution

f(z) = P(Z=z|Z<1) = P(Z=z AND Z < 1) / P(Z < 1).

I thought the top could be simplified to P(Z=z) for z < 1. Correct?

So,

f(z) = 0, for z > or = 1
f(z) = P(Z=z)/P(Z<1), for z < 1.

Correct?

## Answers and Replies

What you have is essentially correct, although it seems bad form to use probabilities in the statement of a probability density function.

I think better is to write: Let g(z) be the PDF of Z = X + Y. If this is still the problem with Z = X + Y where X and Y are uniform on [0,1] that you've been working on in different ways, then you have this. But even if it's not you can just write it in terms of g(z).

So then:

$$f(z)=\left\{\begin{array}{cc}0,&\mbox{ if } z > 1\\ \frac{g(z)}{\int_{-\infty}^1 g(z) dz}, & \mbox{ if } z \leq 1\end{array}\right$$

Last edited:

hgfalling, thank you so much for all the probability help!