Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The input $X$ to a binary communication channel assumes the value $+1$ or $-1$ with probability $\frac{1}{3}$ and $\frac{2}{3}$ respectively. The output of $Y$ of the AWGN channel is given by $Y=X+N$ where $N$ is zero mean Gaussian noise with variance $=1.$

Find the conditional pdf of $Y$ given $X=+1$.

So I started working on it:
I am not sure how to deal with N.

share|cite|improve this question
Since you want the conditional distribution of $Y$ given $X=1$, you don't need to worry about the probability that $X=1$. What is the mean of $Y$ if $X = 1$? What is its variance? What family of distribution does it have? – Jonathan Christensen Oct 29 '12 at 19:50

We are conditioning on $X=1$. So $X$ is $1$. And therefore $Y=1+N$ is just a shifted standard normal, so a normal of mean $1$, variance $1$. The density function, if that's what you want, is therefore $$\frac{1}{\sqrt{2\pi}}e^{-(y-1)^2/2}.$$

share|cite|improve this answer
do we divide by probability of x being 1 which is 1/3? – user46261 Oct 30 '12 at 20:22
No, the conditional density function is precisely the one written above. – André Nicolas Oct 30 '12 at 20:43

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.