Q1. Let x be a random variable with density function {(x + f(x) = (k+1)x² 0
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A: Check the solution in the explanation.Explanation:Check the image below for a detailed solution.
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- 8)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the joint probability density function of the order statistics X(2), X(4), X(6) for n = 7.Let X and Y be continuous random variables with joint probability density function f(x, y) = (3/200y if 0 < 5x < y < 10 O otherwise Find Cov(X, Y)Explain A, B, C
- 7. Let X and Y denote two continuous random variables. Let f(x,y) denote the joint probability density function and fx(x) and fy (y) the marginal probability density functions for X and Y, respectively. Finally let Z = aX + bY, where a and b are non-zero real numbers. (e) Derive an expression for Cov(Z) as a function of Var (X), Var(Y) and Cov(X,Y). [You may use standard results relating to variance and covariance without proof, but these should be clearly stated.]Suppose X is a continuous random variable with density f(x) = x/2 , 0 <= x <=2 f(x) = 0 , elsewhere Write an integral expression for the moment generating function M(t).Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0Let X have a gamma distribution with pdf 1 f(x) Γ(α)βα What is the probability density function of Y = ex? ·xα-¹e-x/B, 0 0, ß > 0Let X be a random variable with density function S (k + 1)r2 0Show that the following are the probability density functions: fi(x) = e-*I(0,c0) (x) f2(x) = 2e¬*I(o,00) f(x) = (0 + 1)f1 (x) – Of2(x) 0 < 0 < 1Let Y1,..., Yn denote a random sample from the density function given by fy (yla, 0) = F(a)0aY"e3 for y >0, where a > 0 is known and r(-) is the gamma function. Find the MLE of 0.6. Let X and Y be continuous random variables with joint density function 24xy if 0 < x < 1,0 < y < 1 – x f (x, y) = 0. otherwise. Calculate E(Y|X = }).Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON