Conditional Probability, Additive Signals

[Anthony45802]

A signal, X, is a random variable with the following density function:
$$f_{X}(x) = \begin{cases}<br /> \frac{3}{25}(x-5)^2, & 0 \le x \le 5\\<br /> 0, & otherwise<br /> \end{cases}$$

The signal is transmitted through an additive Gaussian noise channel, where the Gaussian noise has a mean of 0 and a variance of 4. The signal and noise are independent.

Find an expression for the conditional density function of the signal, given the observation of the output.

Obviously, this is a homework assignment, so I don't want it done for me; however, I am confused. Perhaps I am just confused by the problem or the wording, but I am totally stuck on what to do.

I believe the output signal should be a convolution where Z = X + Y, and Y is the gaussian(0, 2). After I solve the convolution and receive Z, I don't know what to do.

 

[Stephen Tashi]

My guess is that you need to solve for the conditional cumulative distribution $F(w,y) = P(X + Y \le w | Y = y) = P(X \le w - y | Y = y)$. Then differentiate with respect to w to get the density. (I think you interpreted "gaussian noise channel" correctly in the context of the problem, but I think it has an interpretation as a continuous stochastic process in other contexts.)

 

[Stephen Tashi]

If g(X,Y) is joint density of the convolution X + Y, can you set Y = y and integrate symbolically with respect to X from minus infinity to w-y to obtain $P(X \le w -y| Y = y)$? (I haven't worked the details out, so I don't know.)

 

[ssd]

If output (Z)= noise(Y) + signal (X) then the following is the working principle:
Find the distribution of Z.
Then from the joint distribution Z and X find the distribution X|Z.