Let X and Y be two continuous random variables such that the joint density function i given by: f { (a² + y") 0
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- (3) Find the density of Y = X² when X ~ Binomial(4, ). The following answers are proposed: (a) P(Y = k) = ()/2*, k = 0,1, 2, 3, 4. (b) P(Y = k) = (()/2* )", k= 0,1,2,3, 4. (9) %3D %3D %3D (c) P(Y = /F) = k = 0,1, 2, 3, 4. 24 (8) (d) P(Y = k²) = k = 0, 1, 2, 3, 4. 24 (e) None of the above(b) Let X1, X2, ., X, be rvs with pdf 1 f(x\0) = }20 ; -03. If the continuous random vector (X, Y) with conditional density function Y given X = x is f(x)=2e-0¹-*), x 2|X = 1) (c) E[X] and E[X|Y=y].Please solve this question quickly in probabiltyA random variable X has a density function (cx² f (x) = {cx 1 2) P(1/2Let x and y have joint density function * + 2y), for 0 3 Enter the exact answer. (1>}) - 3 1 (b) Find the probability thatx < + y. Enter the exact answer. px +Suppose that X and Y are random variables with the joint density function cx² + cy, 0 sx< 3,1Thank youLet X1, X2,., Xn be a random sample from a uniform distribution on the interval [0, 0] , so that f(x) = 1/0 if 0 s x< 0 Then if Y = max (X), it can be shown that the random variable U = Y/0 has density function f(u) = nun-1 if 0 sus1 If P( (a/2)1/n < Y/0 < (1-a/2)/n)=1-a а. Derive a 100(1-a)% Cl for 0 based on this probability statement. If my waiting time for a morning bus is uniformly distributed and observed waiting times are x,=4.2, x2=3.5, X3=1.7 ,X4=1.2 , and x5=2.4, (Use 3 digits after decimal point) 95% CI for 0 is [ b. If P( a/n < Y/0 < 1)=1-a Derive a 100(1-a)% CI for 0 based on this probability statement. If my waiting time for a morning bus is uniformly distributed and observed waiting times are x1=4.2, x2=3.5, X3=1.7 ,X4=1.2 , and x5=2.4, (Use 3 digits after decimal point) 95% CI for 0 is [ Which of the two intervals derived previously is shorter? C.Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON