Consider the random variable X with PDF f(x) = e−x / (1 + e−x)2 , x ∈ R.
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Consider the random variable X with PDF
f(x) = e−x / (1 + e−x)2 , x ∈ R.
Find the density
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- Two non-negative random variables X and Y have a joint density function f(x, y) = re-*v +1) x > 0, y > 0 Find the correlation coefficient.The number X is chosen at random between 0 and 1. determine the density functions of the random variables V = X/(1 - X) and W = X(1 - X)Be X a random variable with density function defined by f(x) = -2², , x>0. 3 Get the moment-generating function and based on it, calculate the average and variance of X.
- Let X and Y be two independent random variables with densities fX(x) = e^(-x), for x>0 and fY(y) = e^y, for y<0, respectively. Determine the density of X+Y.Suppose you have two independent random variables X ~ Y ~ Exponential(A2), A1 > 0, d2 > 0. Find: Exponential(A1) and (a) The joint density function for (X,Y). (b) P( > 1). (c) P(X +Y > 1).The random variable Y has probability density function fV) = k(y + y®). 0 2. Hence find P(;< Y <). iii) Find the variance of Y.
- Let X be a continuous random variable with a uniform distribution in the interval [0, 1], i.e., X ∼ Uniform(0, 1). We define a new random variable Y as Y = e^X. Find the probability density function (PDF) of Y.Find the expected value and variance of the random variableX whose density function is f(x) = 1/(2(x)^0.5), 1 ≤ x ≤ 4.Let random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.
- Let f(x) is density function of continuous random variable x then the (expected value E(x Ek) = [ xfx)dx Ew) = [_fx)dx - 00 Ek) = ] -xfx)dx Ex)= | xf(x)dxUse the transformation technique to find the density function for the random variable Y. The density function should be given as a piecewise function. f(x) = 1/3e^-1/3x x>0 Y=e^x 0 elsewhereFind the mean and variance for the distribution of random variable X whose density function is f(x). f(x,y)= 1/16x^2e^-x/2 x>0