Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx),...
Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx), where 4 -2542-1 120 0, Find the CDF and PDF of the random variable Y.
Suppose that X is a Gaussian Random Variable with zero mean and unit variance. Let Y=aX3 + b, a > 0 Determine and plot the PDF of Y
Q1. Let X be a random variable uniformly distributed over [-2, 4] (1) Find the mean and variance of X. (2) Let Y 2X+3. Draw the PDF of Y [8 marks] 6 marks] [8 marks (3) Find the mean and variance of Y
Let Y-ar+b (a) Find the mean and variance of Y in terms of the mean and variance of X b) Evaluate the mean and variance ofY if Xhas the following PDF: (a)-ele (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance d) Evaluate the mean and variance of Yif X-bcos 2U) where U is a uniform random variable in of 1 the unit interval. Let Y-ar+b (a) Find the mean...
Let X variable Y by be a normal random variable with mean 0 and variance 1. We define the random y2 if x 20, Y= (a For t E R, compute Mr()-Elen'], the moment generating function of Y. Compute EY
Let \(X\) be a normal random variable with mean \(\mu\), variance \(\sigma^{2}\), pdf$$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} $$and mgf \(M(t)=e^{\mu t+\frac{1}{2} \sigma^{2} t^{2}}\)(a) Prove, by identifying the moment generating function of \(a+b X\), that \(a+b X \sim\) \(N\left(a+b \mu, b^{2} \sigma^{2}\right)\)(b) Prove, by identifying the pdf of \(a+b X\) (via the cdf), that \(a+b X \sim N(a+\) \(\left.b \mu, b^{2} \sigma^{2}\right)\)
Problem1 Let Y=aX + b . (a) Find the mean and variance of Y in terms of the mean and variance of X (b) Evaluate the mean and variance ofYifXhas the following PDF (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance of 1 d) Evaluate the mean and variance of Yif X bcos(2RU) where U is a uniform random variable in the unit interval. Problem1 Let Y=aX + b...
Problem 1. Let X be a normal random variable with mean 0 and variance 1 and let Y be uniform(0.1) with X and Y being independent. Let U-X + Y and V = X-Y. For this problem recall the density for a normal random variable is 2πσ2 (a) Find the joint distribution of U and V (b) Find the marginal distributions of U and V (c) Find Cov(U, V).
Let X be Gaussian with zero mean and unit variance. Let Y = |X|. Find: a) The PDF fY (y) b) The mean E[Y ] c) Here X is uniform in (0, 1), but now you are asked to find a functiong(·) such that the PDF of Y = g(X) is ?2y 0≤y<1fY (y) = 0 otherwise
Let X be a random variable with pdf S 4x3 0 < x <1 Let Y 0 otherwise f(x) = {41 = = (x + 1)2 (a) Find the CDF of X (b) Find the pdf of Y.