The output of a communication channel is Y = X + N where X and N are independent

Question

The output of a communication channel is Y = X + N where X and N are independent, zero mean, random variables. X represents signal and N is noise. (a) Find the correlation coefficient between the input X and the output Y. (b) Suppose we form an estimate of X (call it 'X') by multiplying Y times a constant a or 'X' = aY. Find the value of the constant a that minimizes the mean square error given by E[(X-'X')^2]. (c) Express the resulting (mean square error)/(standard deviation of X)^2 and plot the result. Note: (Standard deviation)^2 is the variance of X. Please do THIS problem clearly for a 5-star rating... NOT a similar one... Thank you!

Explanation / Answer

a)1 b)'X'=E(X)=0 so a=0 c)for that to be minimum it is a straight line with slope 1 and intercept N

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