Ques 3 (d) Suppose that n-10, and Xi Xio represent the waiting times that the 10...
Only ques 4 (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , Xn ~ (iid) Uniform(0,0), E(R)-θ . What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively. 4. Let X. Xn be a random sample from a population with pdf xotherwise Let Xa)<..< X(n) be the order statistics. Show that Xa)/X() and X(n) are independent random variables 5....
3. Again, let XXn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xo) and X(a) (b) Define R-X(n) - Xu) as the sample range. Find the pdf of R (c) It turns out, if Xi, X, n(iid) Uniform(0,e), E(R)- What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively.
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xi) and X(n)- (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , xn (iid) Uniform(0,0), E(R)-θ . What happens to E® as n increases? Briefly explain in words why this makes sense intuitively.
2. Let Xi, X2,...,Xn be independent, uniformly distributed random variables on the interval 0,e (a) Find the pdf of X(), the jth order statistic. b) Use the result from (a) to find E(X)). the mean difference between two successive order statistics (d) Suppose that n- 10, and X.. , Xio represent the waiting times that the n 10 people must wait at a bus stop for their bus to arrive. Interpret the result of (c) in the context of this...
3. Let Xi,... , Xio be a random sample of size 10 from a gamma distribution with α--3 and β 1/e. The prior distribution of θ is a gamma distribution with α-10 and B-2. Recall that the gamma density is given by elsewhere, (a) Find the posterior distribution of θ (b) If we observe 17, use the mean of the posterior distribution to give a point estimate of θ.
1.(c) 2.(a),(b) 5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
+1,20] distribution, where , X.) 3. Let Xi,...,X be a random sample from Uniform θ > 1 is unknown. Let X(1)-min(X, , X.) and X() = max(X,, (a) Derive the edf of X(n) and then its pdf. b) Derive EoX(n) (c) Find a function g(X(n)) such that Eolg(X())-θ for all θ > 1. (d) Replace X(n) by Xu) in the above questions, parts (a) - (c), and answer those.
Let Xi,... , Xn be a random sample from a normal random variable X with E(X) 0 and var(X)-0, i.e., X ~N(0,0) (a) What is the pdf of X? (b) Find the likelihood function, L(0), and the log-likelihood function, e(0) (c) Find the maximun likelihood estimator of θ, θ (d) Is θ unbiased?
3. Let Xi,... , X,n be a random sample from a population with pdf 0, otherwise, where θ > 0. a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ (c) Find the pdf of θ in (b).
5. (10 points) Let Xi, . . . , xn be 1.1.d. from pdf: f(z:0)-D"-9 rS θ, θ > 0, (a) Show that the order statistics (b) Find MLE of θ (1) and X(n) are jointly sufficient for θ.