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- 3) Find the mean and variance of the random variable X with probability function or density f(x). (a)f(x)={k(1-x), –15xs1 O otherwise (b) ƒ(x) = k/3× , x = 1,2,3,... Note: in this problem, you need to first find the values of k.If the random variable X and Y have the joint probability distribution f(x, y) = x+y, 0IF it was XG (23) Independent of Xen G (13) Find Miley D-Probability density Function of the Variable Y-X+Xq 2 -the Mean variance of the variable3. Let Y be the number of speeding tickets a YSU student got last year. Suppose Y has probabilitymass function (PMF)y 0 1 2 3fY (y) 0.12 0.13 0.33 0.42(a) What is the probability a YSU student got exactly one ticket?(b) What is the probability a YSU student got at least one ticket?(c) Compute µY , the mean of Y .(d) Find the variance and standard deviation of Y .(e) What is the probability that Y exceeds its mean value?C1. Let X be a continuous random variable with PDF f(x) = (2-x) ² for -1 1)? (c) Calculate the expectation of X. (d) Calculate the variance of X.The joint PDF of the random variables X and Y is defined as f(x, y) = 25e³"; 0 0 = 0, otherwise %3D (i) (ii) Find the marginal PDFS and X and Y What is the covariance of X and Y?If the joint probability distribution of X and Y is given by x+y f (x, y) =, for x = 0,1,2,3; y = 0,1,2. Find the marginal distribution of Y %3D %3D 30 f(x, y) 1 2 3 1 2 3 30 2 30 30 3 4 Y 30 30 30 30 3 2 30 30 30 30 4.The random variable X has probability function P(X =x)=kx, x =1,2,3, k(x +1), x =4,5, where k is a constant. Find, to 1 decimal place, Var(4 − 3X).C1. Let X be a continuous random variable with PDF f(x) = (2-x) for -1 ≤ x ≤ c and f(x) = 0 otherwise. (a) Explaining your work, find the value of the constant c. (b) What is P(X > 1)? (c) Calculate the expectation of X. (d) Calculate the variance of X.