c) Find the mean and variance of the statistic Y, if i) _Y = 2=1X. ii) Y=1 4X₂² X3 d) if y = X₁² + X3, what is the probability that Y is at most 12.833?
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- X is a discrete random variable and takes the values 0,1 and 2 with probabilities of 1/6, 1/3 and 1/2, respectively. What is the moment generator function M(t) of X?X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2Use the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.
- Let Mx, y be the moment generating function of random variables that are not independent of X and Y. Which of the following / which are not the properties of the function Mx, y?Find E(R) and V (R) for a random variable R whose moment-generating function ismR(t) = e2t(1-3t2)-1Use the moment generating function to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n.Find the distribution of Y = X1 + · · · + Xn.
- Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)X1 and X2 are independent random variables such that Xi has PDF fXi(x)={λiexp(−λix) when x≥0, 0 otherwise}. What is P[X2<X1]?
- If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?A poisson random variables has f(x,3)= 3x e-3÷x! ,x= 0,1.......,∞. find the probabilities for X=0 1 2 3 4 and also find mean and variance from f(x,3).?Find the moment generating function of the continuous random variable X∼U (a, b).