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- 1.3. Let Y₁, Y₂, ..., Yn denote a random sample of size n from a population with a uniform distribution = Y(1) = min(Y₁, Y₂, ..., Yn) as an estimator for 0. Show that ê on the interval (0, 0). Consider is a biased estimator for 8.(a) Given that X is normally distributed with mean 10 and ?(? > 12) = 0.1587. What is the probability that X will fall in the interval (9, 11)? You are given: Φ(1) = 0.8413 and Φ(−1/2)=0.3085, where Φ(?) = ?(−∞ < ? < ?)such that ?~?(0,1). (b) Suppose that during rainy season on a tropical island the length of the shower has an exponential distribution, with parameter ? = 2, time being measured in minutes. What is the probability that a shower will last more than three minutes? If a shower has already lasted for 2 minutes, what is the probability that it will last for at least one more minute?5. Suppose that X1, X2, .. , Xn are an iid sample from the distribution given below, f(X)e = (1/2)(1+ xe); –1 < X < 1. Show that 3X is a consistent estimator of e
- Q. 25 If a Poisson variate X is such that P (x = 1) = P(x=2), Work out P (x = 4).Q. 35 Estimate the parameter λ of the exponential distribution, f(x;^)=λe for 0≤x≤∞o, by the method of moments.Let X1, X2, .., Xn be a sample from a Continuous Unif(u - Ō, µ + d) distribution. Please find the moment estimators (MoM) for µ and ō. [Note] ð > 0. .... (b-a)? [Hint]: If X - Unif (a, b), then E(X)="", V(X)="1" a+l
- State the CLT for independent identically distributed real RVs with mean u and variance a². Suppose that X. X,. . . are independent identically distributed random variables each uniformly distributed over the interval [0, 1]. Calculate the mean and variance of In X. Suppose that 020. Let X1, X2, ables, and set X, be independent, Exp(a)-distributed random vari- Y1 = X(1) and Y = X(k) – X(k-1); for 28. Assume that X is a continuous random variable with pdf 1 if -6Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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