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![Example 17-22. Given the probability density function :
f(x: 0) = [T{1+ (x – 0)}]-1; – ∞ <x< ∞, – ∞ < 0 <∞
show that the Cramer-Rao lower bound of variance of an unbiased estimator of 0 is 2/n, where
n is the size of the random sample from this distribution.
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- 1. Let X be the mean of a random sample of size n = 64 from the exponential distribution e-2/², x>0 otherwise 0, f(x)= = Based on central limit theorem, approximate the probability P (1.8 < X < 2.1). Round your final answer to 4 decimals.(a) Let X1 ~ x² (n, 8) and X2 ~ x² (m, 1). mX1 and its parameters. nX2 (i) Find the distribution of X = (ii) If 1 = 0, name the distribution of X given in 3(a)(i).(a) Let Z follow a standard normal distribution. i. Find the equi-tailed 95% probability interval, i.e. find a > 0 such that P(Z € (-a, a)) = 0.95 and express your result in terms of the inverse function -1 of the standard normal cdf (note that the inverse function satisfies -((x)) = x for all r E R). Finally write down the value of a to three significant digits. ii. Instead of an equi-tailed interval, consider the interval (-a, b) where a, b >0 are such that P(ZE (-a, b)) = 0.95. Express b as a function b(a) of a. It may help to express your results in terms of -1 and . Show that the derivative of the length I of the interval, I = b(a) + a, is given by dl = 1 da (a) (-(0.95+ (-a))' where o denotes the standard normal pdf. Hence obtain a candidate value of a for which the length of the interval is minimal by equating this derivative to zero. You do not need to show that this candidate value is an actual minimizer.
- 1. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).(24) Let X be a random variable with p.d.f. Ze-2 0Question 2: Suppose we have a random sample X₁, X2,..., X, from the following probability density function: f(x;0) 1 - e 20 -1/(20), x>0, 0>0 Derive the maximum likelihood estimator of 0. Be sure to check the second derivative.(a) Let Y be a random variable distributed as X. Determine E(Y) in terms of r. (b) Let {X1, X2, . .. , Xn} be a random sample drawn from a normal distirbution with mean u and 1 variance o?. Denote S E-(X; – X)² as the sample standard deviation. Use the 1 n - result in part (a), or otherwise, to find E(S). (c) Find an unbiased estimator for the population standard deviation o.1. Verify whether the following functions can be considered as probability mass functions: (i) P(x = x) = x² + 1 18 (iii) P(X= x) = -, x = 0, 1, 2, 3 (ii) P(X - x) - ²-2, -, x= 1, 2, 3 2x + 1 18 -, x = 0, 1, 2, 3 [Ans.: Yes) [Ans.: No] [Ans.: No]Suppose X is a discrete random variable which only takes on positive integer values. For the cumulative distribution function associated to X the following values are known: F(23) 0.34 F(29) = =0.38 F(34) 0.42 F(39) 0.47 F(44) = 0.52 F(49) 0.55 F(56) = 0.61 = Determine Pr[29Q1 (a) Given probability distribution function for discrete random variable X as below: 2 4 8 10 P(X = x) 0.08 0.25 0.35 0.2 0.12 (i) Find the mean of X. (ii) Find the variance of X. (b) Let the continuous random variable X has probability density function as follows, (k(x + 2)², -2Question 1 Let X be a continuous random variable taking values in [a, b] with c.d.f. Fx which is strictly increasing on [a, b]. (a) Show that the random variable Fx(X) has a uniform distribution on [0, 1]. (b) Let U be a uniform random variable on [0, 1]. What is the distribution of the random variable F¹(U), where Fx¹ is the inverse of Fx? (c) Suppose that U1, U2, . Un are a set of computer-generated pseudo-random numbers (assumed to be drawn from a uniform distribution on [0, 1]). How would you use them to simulate a random sample X₁, X2,..., Xn from the distribution with density f(x) = = με-μα, x ≥ 0?(b) Consider the generalised three-parameter beta distribution with pdf 121?x (1 – x)² [1- (1 – 2) x]5* fx (x) = 0Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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