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- 3. A random variable X has a cumulative distribution function if 0 0.5). (e) Find P(X < 1.25). (f) Find P(X = 1.25).Let X1 and X2 be IID exponential with parameter > 0. Determine the distribution ofY = X1=(X1 + X2).3. Let X1, and E- X; are independent. , Xn be iid with exponential distribution with mean 0. Show that (X1+X2)/ E=1 X;
- 2. Assume X ~ Exp(0), so that its pdf is 1 f(x; 8) x > 0,0 < 0 < o. e We have 5 samples, 1,2,3,4, and, 5, under Exp(0). Find the maximum likelihood estimator of 0 using the five samples.der a random sample X1, X2, ..., X, having the pdf, f(r; 0) 15. Let X be a positive random variable; i.e., P(X - log(EX) (b) E[log(1/X)] > log[1/EX] (c) E (X³) > (EX)³Let x1, x2, ..., n represent a random sample from a distribution with mean E(x) and variance Var(x). Show that Cov(x, x₁ - x) = 0.#5. Two dice are rolled 300 times. Let X = number of double sixes obtained (out of 300 rolls). (a) Find P(X = 8) using the Poisson approximation to the binomial. (b) Find the mgf of W = 3X + 2.(b) Let X represent the lifetime in months of a certain brand of light bulb. The pdf of X is given by f(x)=K(6-x), 0The time to wait between each phonecall a person recives is random given f(t) = 3e-3t for t>=0. Let T1 and T2 be two independent waiting times for this distribution. Find expected time between each call and variance. (E(T) and V(T)) Find the probability for both T1 and T2 to be greater than one.Q1 Let X1, .. X, f(r;0) = (;)*"(1 – 0)-,1 = 0, 1,2. %3D (i) Find the score and information functions of 0. (i) Show that X is an efficient estimator 0. إضافة ملف3. (20%) Let W1 < W2 < W3 < W4 < W5 < W6 be the order statistics of n = 6 independent observations from f(x) = x/2; 0 < x < 2. (a) Find E(W6)SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON