Let X and Y be independent random variables each having the uniform distribution on [0, 1]. Let U = min{X, Y) and V = max(X, Y). Find E(U), and hence calculate cov(U, V).
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![Let X and Y be independent random variables each having the uniform distribution on [0, 1]. Let
U = min{X, Y) and V = max {X, Y). Find E(U), and hence calculate cov(U, V).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f55cdfd-3542-42d7-825e-e0a092b1c59a%2F8b9523ea-c182-49a0-9dca-725804af4f9b%2F2cqwgyi_processed.jpeg&w=3840&q=75)
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- Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of YLet X and Y be random variables having the same distribution. Show that Cov(X +Y, X – Y) = 0.Let X1, X2, · · · , Xn be uniform i.i.d. random variables between [0, 1].Find the CDF and PDF of X = min(X1, X2, · · · , Xn)
- Let X and Y be two jointly continuous random variables. Let Z = X² + Y². Show that F2(=) = L Fxr(V=- yř 19) – Fxy(-v/= = y"l»)] fv (w) dy .Let random variable X be uniform in the interval (0, 1). Define random variable Y = aX + b where a not 0.The 4-dimensional random vector X has PDF fX(x)={1 when 0≤xi≤1, i=1,2,3,4 and 0 otherwise}. Are the for components of X independent random variables?
- The random vector x obtained by adding the vectors x1, x2 and x3 end-to-end has a multivariate Gaussian distribution. How do you find the distribution of x2?Q4) The probability mass function of Y is f(y) = y/6 for y=1,2,3,4. Find the variance of Y.Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)
- Assume that X and Y are independent random variables where X has a pdf given by fx(x) = 2aI(0,1)(x) and Y has a pdf given by fy(y) = 2(1– y)I(0,1)(y). Find the distribution of X + Y.Prove that cov(X, Y) = cov(Y, X) for both discreteand continuous random variables X and Y.Let the random variable X fit uniformly in the interval (-1,1). Find the distribution of the random variable X².
