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![B2. Let X and Y be two continuous random variables with joint density function
S cr*y, if 0 <y SI<1,
fx.x(x, y)
1 0,
otherwise.
(a) Find the value of constant c.
(b) Find the marginal density functions fx(x) and fy (y). Are X and Y independent?
why/why not?
(c) Find the conditional density fyx=z(y|x), use this to compute E[Y|X], and show
that E[E[Y|X]] = E[Y].
(d) Find P(Y <).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2301df6-3a95-4265-ae4d-aabbf03d3289%2F086941bb-6c8d-4fd7-a606-75518a5a8b4a%2Fwuej3v_processed.jpeg&w=3840&q=75)
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- Let X1, X2, . X6 be an i.i.d. random sample where each X, is a continuous random variable with probability density function f(x) = e-(-0) , x > 0 Find the probability density function for X(6).The multidimensional random variables H= (c) Let x := 2731 W 21 X2 Yı = follows N₁(μ, E) where 7 3 Y 2 3 Y 6 0 0 5 4 -2 2 4 -2 G (a) Determine all the independent univariate random variables. (b) What is the probability density function for x₁ + 2x₂? [2] tion density function p(y|x) and p(x|y = 1). and y := [₁]. Determine the conditional distribu-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.]
- Exercise 40 Let X Unif(0,1). Let g(x) = e" and Y = g(X). (i) Find the density fx of the random variable X. (ii) Find the cdf Fy (y) = P(Y < y) of Y. (iii) Find the density fy of the random variable Y.2. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 0b) Let X and Y be jointly continuous random variables with the following joint density function (1/32)xy ,0 2) v. Evaluate P(Y > X – 2)Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0X and Y re random losses with joint density funcTion 2C forocn<00 500000 An Insurance losses pays the total of the twX9 losses to a maximum and oey< lo0, 100 policy on the payment of 100 the expectic fpagment the insurer will make on this policy.he joint probability density function of two random variables X and Y given by fxY (x, y) = c (2x +y) for 0 SxS1, 0S y S2 elsewhere Find (a) the value of C4) The joint density function of the random variables X and Y is given by (&xy f(x, y) = 0SXS1,0 sysx otherwise Find (a) the marginal density of X, (b) the marginal density of Y, (c) the conditional density of X, (d) the conditional density of Y.Let X and Y have the joint probability density function fxy(x, y) = {#², for 0Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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