Question 1: Suppose that X1, X2,... Xn are independent identically distributed continuous outcome random variables which...
Let X1, , X2 ... be a sequence of independent and identically distributed continuous random variables. Say that a peak occurs at time n if Xn-1 < Xn < Xn+1 . Argue that the proportion of time that a peak occurs is, with probability 1, equal to 1/3
Consider n independent and identically distributed random variables X1,X2, following a uniform distribution on the interval [0,1] ,Xn, each a) What is the pdf of Mmin(X1,X2, .. ,Xn)? b) Give the expectation and variance of XX 1-1лі.
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint probability density function: fX,Y (x,y) = e-y, 0 <x<y< ; =0 , elsewhere. Evaluate P( X2>X1, Y2>Y1) + P (X2 <X1, Y2<Y1) .
Suppose X1, X2, ..., Xn are independent and identically distributed (iid) with a Uniform -0,0 distri- bution for some unknown e > 0, i.e., the Xi's have pdf Suppose X1, X2,..., Xn are independent and identically distributed (iid f(3) = S 20, if –0 < x < 0; 20 0, otherwise. (a) (4 pts) Briefly explain why or why not this is an exponential family (b) (5 pts) Find one meaningful sufficient statistic for 0. (By "meaningful”, I mean it...
2. Let X1, X2,. . , Xn denote independent and identically distributed random variables with variance σ2, which of the following is sufficient to conclude that the estimator T f(Xi, , Xn) of a parameter 6 is consistent (fully justify your answer): (a) Var(T) (b) E(T) (n-1) and Var(T) (c) E(T) 6. (d) E(T) θ and Var(T)-g2. 72 121
(a) Suppose that X1, X2,... are independent and identically distributed random variables each taking the value 1 with probability p and the value -1 with probability 1-p. For n = Yn-X1 + X2 + . . . + Xn. Is {Y, a Markov chain? If so, write down its state space and transition probability matrix 1, 2, . . ., denne
15. Let X,, X2,.. . be independent, identically distributed random variables, EIXI oo, and denote S,-X1+... + Xn. Prove that [Use symmetry in the final step.] 15. Let X,, X2,.. . be independent, identically distributed random variables, EIXI oo, and denote S,-X1+... + Xn. Prove that [Use symmetry in the final step.]
Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables with common probability function ()p(1-p) m m-a ; x 0,1,. ., m otherwise 0 where m is known and p is unknown (a) Obtain the Sequential Probability Ratio Test of Ho p = po versus HA p P, where pi > po, with significance level 0.01 and power 0.95. Describe the test precisely; (b) For the case where po 3/8,pı = 1/2, m =...
4. Let X1,..., Xn be independent, identically distributed random vari- ables with common density 2 log c)? f(0; 1) = 0<<1, XCV21 (>0). : 212 (a) Find the form of the critical region C'* for the most powerful test of H:/= 1 vs. HQ: >1. (b) Suppose the n = 20 and a = .10. Find the specific value for the cutoff value) K from the critical region C* in part (a). (Hint: Show that Y = (log X/X) is...
Let X1,X2,...,Xn denote independent and identically distributed random variables with variance 2. Which of the following is sucient to conclude that the estimator T = f(X1,...,Xn) of a parameter ✓ is consistent (fully justify your answer): (a) Var(T)= (b) E(T)= and Var(T)= . (c) E(T)=. (d) E(T)= and Var(T)= We were unable to transcribe this imageWe were unable to transcribe this imageoe We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...