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![3. Suppose that X and Y have the joint probability density function
f(x,y) = {y-¹e-, if y>0 and 0<x<y
0,
otherwise
Find E[X], E[Y], and COV(X,Y).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F025e6abb-25a8-4668-a3cf-4209699125ab%2F16fb4323-fe6d-47c3-a6d1-f44f7172c46d%2Fxal0xjh_processed.png&w=3840&q=75)
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- 3. Suppose that X and Y are independent with X ~ N(0, 1) and Y~ N(0,4). Let U = X + Y and V = X - Y. a) Find the joint probability density function of (U, V), fu,v(u, v). b) Are U and V independent?Help me pleaseLet 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< 3,12. Let Y₁, Y2,..., Yn denote a random sample from the probability density function (0+1)yº; 0 −1 otherwise. f(y;0) = { 0, (a) Find the MLE for 0, say . Be sure to check the second derivative. (b) Find the method of moments estimator of 0, say . (c) Suppose that 1000 'yi = 750.7516 and 1000In(y) = -329.0056, calculate 8 and 0. i=1Suppose 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).The lifetimes X and Y of two components in a machine have the joint density function f(x, y) = (1/4)*(2y + 2 − x) for 0 < x < 2, 0 < y < 1 and f(x, y) = 0 otherwise. (a) What is the probability density of the time until neither of two components is still working? (b) What is the probability distribution of the amount of time that the lifetime X survives the lifetime Y? KINDLY REQUEST YOU TO PROVIDE ME WITH COMPLETE SOLUTION4. Let F(x) = S f(z)dz where f(x) is a valid probability density function. Find an expression for d F'(x) = F(x) in terms of the function f. Hint: Look up the Fundamental Theorem of Calculus on Wikipedia and read the corollary.Suppose the life span of a laboratory animal is defined by a Rayleigh density function with a=0 and b=30 weeks: (x-a)² bu(x-a) Where £x (x)= = b (x-a)e 9 = { 1 u(x) = x>0 x < 0 What is the probability that the animal will live between 10 and 20 weeks? What is the expected lifespan of the animal?Let X have the probability density function given by Se-*, 0The joint probability density is for X and Y are f(x, y). Give the exact answer. S 2 (x + y) 0 < x < 1, 0 < y < 1, y < x f (x, y)= elsewhere P (x < }, y < })=. Р (х 2, y < )-Now, assume that Y₁,..., Yn distributed iid, each with density -{* where is positive constant. Find the density function for fy) (y). fy(y) = e-(-0), 0, if y> 0; otherwise,SEE MORE QUESTIONSRecommended 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