Communications ...

Consider a function \({f_X}\left( x \right) = \frac{k}{{{x^2}}}u\left( {x - k} \right)\) where u(x) is the unit step function.

This function will be a valid probability density function, for

Let \(f\left( x \right) = \frac{1}{{\sqrt {\pi \left( 8 \right)} }}{e^{ - \frac{{{x^2}}}{8}}}\) be a random variable function. The function is shifted in time such that the maximum value of function occurs at x = 2. Find the variance of shifted function.

Consider the two random variables X and Y related as Y = X². If the probability density function of X has even symmetry, then

Consider a random variable θ  distributed uniformly over the domain [-π, π]. What is the variance of cos²θ?