Suppose that X and Y are statistically independent and identically distributed uniform random variables on (0,1).(a) Write down the joint probability density function fx,y(x,y) of X and Y on its support.(b) The expression for the joint probability density function of the transformed random variables U=5X+Y and V=3X+2 Y on its support is:fu,v(u,v) = Au³ (C + D)EWhich values of the constants A, B, C, D, E are correct (in the same order as they appear here)?○ 1,8,1,8,1○ none of the answers is correctThe joint probability density is constant on its support, hence B-C=0 and D=E=1, and A following from normalization.○ 1/3, 1, 1.67, 1, -2O 1/3, 1, 1.67, 1, 23, 1, 1.67, 1, -2

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Suppose that X and Y are statistically independent and identically distributed uniform random variables on (0,1). (a) Write down the joint probability density function fxy(x,y) of X and Y on its support. (b) The expression for the joint probability density function of the transformed random variables U = 6 X + Y and V = 6 X + 2 Y on its support is: fulu,v) = A u® (C v + D)E Which values of the constants A, B, C, D, E are correct (in the same order as they appear here)? O 1, 2, 3, 4, 5 O 1/6, 1, 1.00, 1, -2 о 1/6, 1, 1.00, 1, 2 О 6, 1, 1.00, 1, -2 O 17,0,0, 1, 1 none of these answers is correct.
Suppose that X and Y are statistically independent and identically distributed uniform random variables on (0,1). (a) Write down the joint probability density function fx,y(x,y) of X and Y on its support. Answer: (b) The expression for the joint probability density function of the transformed random variables U = 10 X + Y and V = 6 X + 2Y on its support is: fu,v(u,v) = A B (Cv+ D)E Which values of the constants A, B, C, D, E are correct (in the same order as they appear here)? O 17,0,0, 1, 1 O 1/6, 1, 1.67, 1, -2 O 2, 4, 6, 8, 10 O 1/6, 1, 1.67, 1, 2 O none of these answers is correct. O 6, 1, 1.67, 1, -2
Suppose that X and Y are statistically independent and identically distributed uniform random variables on (0,1). (a) Write down the joint probability density function fxy(x,y) of X and Y on its support. Answer: 1 (b) The expression for the joint probability density function of the transformed random variables U=5 X + Y and V=8X+2 Y on its support is: fu,v(u, v)=A³ (Cv+ D) E Which values of the constants A, B, C, D, E are correct (in the same order as they appear here)? O 17,0,0, 1, 1 O 2, 4, 6, 8, 10 O 8, 1, 0.63, 1, -2 O none of these answers is correct. O 1/8, 1, 0.63, 1, 2 O 1/8, 1, 0.63, 1, -2
Let X1, X2, and X3 be independent random variables from (-1, 1). Find the probability density function and the expected value of the random variable [X(1) + X(2)]/2.
Let X and Y be independent normally distributed random variables with mean zero and variances og = 1 and of = 4. (a) Write the joint probability density function fx.y (r, y). • (b) Define new random variables U = aX + Y and V = X – Y, where a + -1 is a real number. Find the absolute value of the Jacobian of transform from X, Y to U, V. (c) Find the joint probability density function for U and V. Find a for which U and V are independent random variables. Write down fu,v (u, v) for this a in the answer.
Let Xand Y be two continuous random variables with joint probability density [3x function given by: f(x.y)%D 0sysxsl elsewhere with E(X) = ECX)- EC) - EC*)= ;and E(XY) = 10 3 E(Y*) = - and E(XY) =; %3D Then the value of the variance of 2X+Y is: O 3/80 O 91/320 43/320 7/20
Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.
Let X be a random variable with density function - 1< x < 2, fn) = fx*/3, f(x) = 0, elsewhere. Find the expected value and the covariance of g(X) = 4X + 3.
Suppose X and Y are two independent Uniform(0, 1) random variables. Use the cumulative distribution function method to find the probability density function of their sum U = X + Y.
Suppose two independent continuous random variables X, Y have exponential distribution with mean 0. a) Write the joint probability density function. b) Express F (2) = P( < z) using integrals and compute the integrals. c) Calculate the PDF of Z = Y/X as f(x) = F' (2) d) Compute the expected value and the variance of Z, if they exist.
Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.
b) A random variable Y has a probability density function defined as: -y 1 f(y)= 0 = e

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