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- Let ZZ be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that Var(Z)=(k+1)E(Z)Var(Z)=(k+1)E(Z) where kk is the last non-zero digit of your student ID number. Determine the distribution of ZZ and find its parameter(s), explaining your argument carefully.Q3) Let X1,X2,. X, be a random sample from the truncated exponential distribution with pdf: f(x,0) = Please else (a) Derive the method of moment estimator of 0; (b) Derive the MLE of 0. (c) Are the MOME and the MLE unbiased estimators for 0? (d) Compare the MSE of the MOME and the MLE for 0. Which one is a better estimator for 01 The PDF of the random variable X is fx(x)= 6√x What are i) mean, ii) the mean square value and iii) the variance of random variable x. CS Scanned with CamScanner 0Suppose we have random samples (X₁, X2, X3, X₁, X5) and it is known that X; Exponential (0). Th observations are (x1, X2, X3, X4, X5)=(0.32, 2.37, 1.26, 2.18, 1.8) When typing in formulae, use the notation below xi theta x_i When typing in formulae, don't forget to use "*" for multiplication Likelihood Find the likelihood function by filling in the gaps below. Express the Likelihood functions through and X1, X2, X3, X4, X5. 5 = The formula for calculating the likelihood is L(0; x) = [] ƒ(X; ; 0); i=1Prob. 3 Let X be a random variable with cumulative distribution function (cdf) given by (1-e-x², x ≥ 0 ={1,- x<0 Find the probability that the random variable X falls within one standard deviation of its mean. Fx (x) =Let X1,...,Xn be a random sample from the distribution f(x) = 2x, 0 < x < 1. Find the distribution of the sample maximum X(n)