6. The joint probability density function of X and Y is given by f(x, y) = .0Y). (c) Find P(Y > }|X < }). (d) Find E[X].
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![6. The joint probability density function of X and Y is given by
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f(z,v) = (-
÷(² +).0 < z < 1,0 < y < 2
(a) Compute the marginal density of X.
(b) Find P(X > Y).
(c) Find P(Y > |X < }).
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- Let x be a continuous random variable with density function +K 9. 0Q1. Let x be a random variable with density function 2 (k+1)x 0 < x < 1 0 otherwise 9 = f(x): Find the moment generating function of x.Suppose that X, Y are jointly continuous with joint probability density function f (x, y) = ce ² (² 2 x, y = (-∞0, ∞0), for some constant c. (a) Find the value of the constant c. (b) Find the marginal density functions of X and Y.The joint probability density function of continuous random variables X and Y is given by J3x, 0 0.5).The cumulative distribution function of the continuous uniform distribution between con- stants a and b is given by F(x) = P(X ≤ x) = x-a - a x b (a) The probability density function is f(x) = F(r). Find the form of f(x). (b) Find the derivative of f(x) for x = [a, b]. Is f(x) decreasing, increasing or flat in this region? (c) Does f(x) have a single maximum in the region [a,b]? If so, what is it, or if not, why not?A random variable X has a density function (cx² f (x) = {cx 1 2) P(1/2Consider two continuous random variables X and Y with marginal distributions g(x) and h(y)respectively and the joint density function given by: x > 0, 0 < y < 2 elsewhere. f (r.y) Then: X and Y ar statistically dependent f(x,y)#g(x)h(y) None of these f(ylx)=g(x)Let A > 0. Suppose (X, Y) has joint probability density function given, for all x, y E R, by 3. fx,y(x, y) = { Ay exp(-y(x+ A)), if a 2 0 and y > 0 0, otherwise. (i) Calculate the marginal density of Y. (ii) Let y > 0. Find the the conditional probability density function of X given Y = y. Interpret your answer. (iii) Calculate E[X|Y] and prove that E[X] = +oo. Hint: You can use without a proof that -dy = +oo. %3DPlease helpLet X Unif[-3, 3], and Y = (X + 1)². (a) Find fy(1) and fy(9), where fy(•) is the probability density function of Y. Separate your answers with a comma. (b) Find P{Y < 9}.Let X and Y have the joint probability density function fxy(x, y) = {#², for 04. Let X, Y be jointly continuous with density f(x, y) = = c(x - y)e-2 File Preview (a) Find the value of c. (b) Find the conditional distribution of X|Y. 0 < x<∞, -x < y < x otherwiseRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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