Consider two random variables X and Y with the following joint density function f(x,y) = {e^** a) Find marginal density function of Y. b) Find f(x| Y = 2) c) Find F(x|Y) 0 ≤ y ≤x<∞0 elsewhere
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- Let Y1 and Y2 be random variables with joint density functionf(y1, y2) = (6/7(y1^2+y1y2/2) 0 < y1 < 1, 0 < y2 < 2,0, elsewherea) Find marginal density functions. Are Y1 and Y2 independent?b) Find P(0 < Y1 < 0.3, −2 < Y2 < 1).c) Find P(0.6 < Y1 < 1|0 < Y2 < 1).b) Suppose that the joint density function of random variables X₁ and X₂ is given by f(x₁, x₂)= = 0, 0< x₁ < x₂ <2 otherwise i) Draw a sketch of the area where f(x₁, x₂) is not zero. ii) Find P(X₂ < 1). iii) Find E(X₁X₂).6) The random variables X and Y have joint density f( x, y) = x2-y2 for 1< x, 1s y and f( x, y) = 0 otherwise. %3! Compute the joint density of U = X /Y and V = X Y.
- 1) The joint probability density function of the X and y random variables, fx,y (xiy)= cx,0Let (X,Y)' have density for r, y > 0, f (x, y) = (1+x)² (1+xy) > 0, otherwise. Show that X and X Y are independent, equidistriduted random variables and determine their distribution.The random variable Y has probability density function f(V) = k(y + y³), 0 2. Hence find PG < Y <). iii) Find the variance of Y.Let X and Y be two continuous random variables with joint probability density function: 6e-(2x+3y)+k x21, y > 0 fxy (x, y) = otherwise a) Find the coefficient k. b) Find P(X 2). c) Find the marginal probability density functions of X and Y ( fx(x), f;(y)). d) Are X and Y independent?5. Let f(x,y) is a joint probability density function of (X,Y) continuous random variable as below. (6 — х — у) 0Let Y1 and Y2 be random variables with joint density function (yỉ + 2), 0 < y1 < 1,0 < y2 < 2, f(y1, Y2) elsewhere a) Find marginal density functions. Are Y1 and Y2 independent? b) Find P(0 < Y, < 0.3, –2 < Y2 < 1). c) Find P(0.6Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0The density function of two random variables X and Y is ,-2(x+y) fx,r (x, y) =u(x)u(y)4e¯¾x*y) X,Y Find the mean value of the function e-*+),1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.SEE MORE QUESTIONSRecommended 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