Let X be a random variable, on [0, 1], with probability density function 1 p(x) = x² + 2 + 3¹3 Let Y be a random variable on [2, 3], such that Y = X² +2. Find the probability density function for Y.
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- Let X be a continuous random variable with density functionf(x) = 3x^-4, x ≥ 1. Compute E(X ) and Var(X ).Suppose that X, Y are jointly continuous with joint probability density function f( x, y){ xe^-x(1+y), ifx >0 and y >00, otherwise. (a) Find the marginal density functions of X and Y. (b) Calculate the expectation E[XY]. (c) Calculate the expectation EIX/(1+ Y )1. (e) Determine if the random variables X and Y in this exercise are independent.Suppose that X is a continuous random variable with density function f(x). If f(x)=k for −5≤x≤3 and f(x)=0 otherwise, determine the value of k.
- Let X and Y be independent uniform random variables on (0, 1). Find their joint density function f (x, y). Use the joint density function to calculate the probability P(X < Y).Let X and Y be two independent random variables, X ∼ Γ(α, λ) and Y ∼ Γ(β, λ). Find the joint probability density function f(Z,W)of the vector (Z, W)(b) Show that Z and W are independent(c) Show that Z ∼ Γ(α + β, λ) and W ∼ B(α, β)Let x be a random variable defined by density function fx=3x^2 define less than equal to 1 and great than equal to 0 and 0 is defined by otherwise. 1) find E(X) 2) Find E(X^2)
- Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise Find the constant c, P(Y≥1/2), P(X < 2, Y >1/2), P(X < 1), Determine whether X and Y are independent.Suppose that the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.
- Find a value of k that will make f a probability density function on the indicated interval. ƒ(x) = kx; [1, 5]Let X and Y be two continuous random variables with joint probability density function f(x,y) = k(x + y) for 0 < x < 1, 0 < y < 1 and 0 otherwise, where k is a constant. Find the marginal probability density functions of X and Y.Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independent