First of all we have to find joint density function
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Suppose that X1,..., X, are a random sample from a distribution with density f(x). Prove that...
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi , . . . , X,.), V=min(X1, ,X,). (a) Find the distribution function and the density function of U and of V (b) Show that the joint density function of U and V is fe,y(u, u)= n(n-1)/(u)/(v)[F(v)-F(u)]n-1, ifu < u. (7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi...
(2) Suppose that X1,..., X, is a random sample drawn from a distribution with the pdf f(x) = 3.c- for 1<x and that Yi <... <Yg are the corresponding order statistics. (a) What is the cdf of Yı? (b) What is the pdf of Yo?
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n = 2m+1 for some integer m. Let Y be the sample median and Z = (a) Apply the usual formula for the density of an order statistic to show the density max(X1) be the sample maximum. of Y is 0) 6 3) (b) Note that a beta random variable X has density re+ β22 a-1 (1-2)8-1 with mean μ α/G + β) and variance...
3. Let X, X,. X, be a random sample from a distribution with density f(x)- ) if 0 < x < θ 0 otherwise (a) (b) Determine the density of =max(X1,X2, ,Xn} Use the result of (a) to show that θ is a biased estimator of θ. Determine the bias, B(6) Calculate the mean-square error, MSE(6). (c)
LetXX X be a random sample from a distribution with density otherwise Determine the density of)-max(X1,X (b) (a) ,&J Use the result of (a)toshow that θ is a biased estimator of. Determine the bias,B( (c) Calculate the mean-square error, MSE(θ)
1. Suppose that X1, X2,..., X, is a random sample from an Exponential distribution with the following pdf f(x) = 6, x>0. Let X (1) = min{X1, X2, ... , Xn}. Consider the following two estimators for 0: 0 =nX) and 6, =Ỹ. (a) Show that ő, is an unbiased estimator of 0. (b) Find the relative efficiency of ô, to ô2.
5.2.5 5.2.5. Let X1, . . ., X, be a random sample from the truncated exponential distribution with pdf f(x)=e-a-0) 0, S otherwise. Find the method of moments estimate of 0. 5.2.5. Let X1, . . ., X, be a random sample from the truncated exponential distribution with pdf f(x)=e-a-0) 0, S otherwise. Find the method of moments estimate of 0.
6. Suppose that X1, ..., Xn is a random sample from a population with the probability density function f(x;0), 0 E N. In this case, the esti- mator ÔLSE = arg min (X; – 6)? n DES2 i=1 is called the least square estimator of Ô. Now, suppose that X1, ..., Xn is a random sample from N(u, 1), u E R. Prove that the least square estimator of u is the same as maximum likelihood estimator of u.
Let X and Y denote independent random variables with respective probability density functions, f(x) = 2x, 0<x<1 (zero otherwise), and g(y) = 3y2, 0<y<1 (zero otherwise). Let U = min(X,Y), and V = max(X,Y). Find the joint pdf of U and V.
Problem 3. Consider a random sample X1, X2,..., Xn from a distribution with log-normal pdf (density function): for t 0 and 0 otherwise. Both μ and σ 0 are unknown parameters. Find the method of moments estinates μ and σ. Hint: computing moments, change of variable y = Int might be useful.