Question: Two points are selected randomly on a line of length
Two points are selected randomly on a line of length L so as to be on opposite sides of the midpoint of the line. [In other words, the two points X and Y are independent random variables such that X is uniformly distributed over (0, L/2) and Y is uniformly distributed over (L/2, L).] Find the probability that the distance between the two points is greater than L/3.
Answer to relevant QuestionsShow that f (x, y) = 1/x, 0 < y < x < 1, is a joint density function. Assuming that f is the joint density function of X,Y, find (a) The marginal density of Y; (b) The marginal density of X; (c) E[X]; (d) E[Y]. The random variables X and Y have joint density function f (x, y) = 12xy(1 − x) 0 < x < 1, 0 < y < 1 and equal to 0 otherwise. (a) Are X and Y independent? (b) Find E[X]. (c) Find E[Y]. (d) Find Var(X). (e) Find Var(Y). Jill’s bowling scores are approximately normally distributed with mean 170 and standard deviation 20, while Jack’s scores are approximately normally distributed with mean 160 and standard deviation 15. If Jack and Jill ...The joint density function of X and Y is given by f (x, y) = xe−x(y+1) x > 0, y > 0 (a) Find the conditional density of X, given Y = y, and that of Y, given X = x. (b) Find the density function of Z = XY. Let X and Y denote the coordinates of a point uniformly chosen in the circle of radius 1 centered at the origin. That is, their joint density is f(x, y) = 1/π x2 + y2 ≤ 1 Find the joint density function of the polar ...
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