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- Let the random variable X have the density function is given by сх, 0If X is continuous random variable then the first moment about the origin is defined to be E (X) = Jxf(x)dx ylgn ihiThe discrete random variable X has probability mass function kx x = 2,3 2kx P(X = x) = x = 4,5 x-1' 0,otherwise. Find the variance of X.A random variable Y have a distribution with parameter a > 0 and yo > 0 if its density function is given by: ´ayf f(y) = {ya+1 (0, if y > yo elsewhere ii) ayg (a-2)(a-1)2" Show whether or not that the variance of Y isConsider the probability density function P. (x) = a e-b lxl where X is the random variable which assum es all the values from - ∞ to o. Find CDr ) 4.1. A continuous random variable X has a probability density function (PDF) p(x) = k( 8-x^2/2) on the interval [0, 4] (a) Find k such that p(x) is a valid PDF. (b) Find P(X ≤ 1). (c) Find the mean, µ, of X. (d) Find the variance, σ 2 , of X.Many casinos have a game called the Big Six Money Wheel, which has 54 slots in which are displayed a Joker, the casino logo, and various dollar amounts, as shown in the table at the top of the next column. Players may bet on the Joker, the casino logo, or one or more dollar denominations. The wheel is spun and if the wheel stops on the same place as the player's bet, the player wins that amount for each dollar bet. Denomination Number of slots $40 (Joker) 1 $40 (Casino logo) 1 $20 2 $10 4 $5 7 $2 15 $1 24 If a player bets $4 on the $10 denomination, find the player's expectation. (Round your answer to two decimal places.)If X, and X, are two random variables having joint density function 2. f(x,,x, ) = +8x,x,* ,0Sx, S 2,0s x,sI 8. Find i. P(X, I) P(X, <1/ X, <0.5) P(X,+ X, <1) ii. iii. iv.A continuous random variable X has a P.D.F. fx (x) = 3 X, 0Sr a) and (ii) P (x > b) = 0.05.