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- 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 0Let X1, . . . , Xn be iid from Exp(β). Find the distribution of the sample maximum Y = X(n). Is this an exponential distribution? (This question was rejected this morning. I did double check and there is no missing info with this question.)Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
- 7. Let X be a continuous random variable with pdf f(x)=√ 0Exercise 10. For an exponential random variable X (Definition 10), show that E[X] = }. More generally, for a function o : R R, E(¢(X)] = | (u) dF(u) (5.4)If Y has a binomial distribution with n trials and a probability of success p, show that the functiongenerator of moments for Y is: m(t) = (pe' +q)n with q = 1− pSuppose the inter-arrival time of people using the elevator can be modeled as X ~ Exp(1/6) (in minutes). What is the distribution of the number of people waiting at an elevator in two minutes (since the last elevator left)? Hint: Let Tn be the time when the n-th person arrives at the elevator, and let N be the number of people waiting at time t, then P(N>=n) if and only P(Tn =n) and P(N>=n+1). The difference of the two gives you P(N=n).The p.d.f. of a random variable X' is as shown in the figure. The pdf is zero for X 5. Calculate (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X. fx (x) kLet X ~ F(a, ß). If E(X)= % and E(X²)= a+a² If â = nX? find the method of (n-1)s² | .3 moments estimator of21. Let the random variable X have the moment generating function e3t M(t) : -1Suppose a businessman wants to determine the monthly income of a new restaurant is 1 a random variable X with the moment generating function Mx(t) = (1-500t)5. i. Compute the mean of X ii. Variance of X iii. Standard deviation of XIf a random variable X has pdf ?(?) = 3/376x(1−?), 0 < ?< 1 and y = √(x+1) what is the pdf of Y? (include the domain)SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,
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