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- Let X and Y be independent exponential random variables with parameter 1. Find the cumulativedistribution function of Z = X/(X + Y )Consider a standard Normal random variableZ ∼ N(0,1). Use the method of cumulative distribution functions, or any other method, to find the probability density function fU (u) of U=Z^5- The Pdf of a random variable X is given by fx (x) = K_ for aIdentify the distributions of the random variables with the moment- generating functions shown below. For each random variable also indicate what the mean and the variance are. a) m(t) = e^2.2(e^t−1)b) m(t) = 1/(1 − 2t)^2 c) m(t) =( e^5t−e^t )/4ttwo random independant variables X and Y with distributions: X ∼ Poisson(λ) og Y ∼ Poisson(2λ), andobservations x = 2 og y = 5 .(a) What is the log-likelihood function? b) calculate MLE for the observated samples5 Both K and M has standard normal distribution and they are independent random variables. Z=K+M Find E[K(K^2+M)^(1/4)].Without using a moment generating function; Prove that the variance of a beta-distributed random variable with parameters α and β is σ2 = αβ/[(α + β)^2 (α + β + 1)]With reference to Definition 4, show that μ0 = 1and that μ1 = 0 for any random variable for whichE(X) exists.Suppose random variable Y represents a random number generator that can take any value between 1 and 11. Then the density curve of the outcomes has constant height between 1 and 11, and height of 0 elsewhere. What’s the height of the density curve between 1 and 11? What is the probability that Y<2? What is the probability that Y=2? Can Y=2 happen as an event for this random number generator? What is the probability that Y=0? Can Y=0 happen as an event for this random number generator?Let X1, X2,...,X, be a random sample from a distribution with density function e if x > 0 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?7. Let X~ N (0,0²) and {X; : i = 1,2,..., n} be a random sample from X. (a) Formulate the log-likelihood function. (b) Find the ML estimator of o². (c) Derive the variance of the ML estimator of o2, 62. Does the variance of ô2 achieve the CR bound? (d) Derive the asymptotic distribution of √n (-o).Suppose that the Cumulative distribution function (CDF) of the random variable X with parameter 0 > 0 is defined by F(x;0) = 1 - e , x > 0 i) What is the expected value of X,E(X)? ii) What is the variance of X,V (X)?