![6. (15 pts.) Let X,X.. Xn be independent and identically distributed erponentially distribu random variables, each with mean ux 1. Let a. Calculate E[W] b, Calculate ơw, the variance of W c. Calculate the probability P[X, Ss 1 d. Approximate the probability P[W 1] when n is large e. Suppose n - 1000, and you have one guess at what W is. What numerical value would you pick? What fundamental result in probability theory are you basing your answer on?](http://img.homeworklib.com/questions/abe64540-fa42-11eb-b8eb-35af36d4f1b2.png?x-oss-process=image/resize,w_560)
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6. (15 pts.) Let X,X.. Xn be independent and identically distributed erponentially distribu random variables, each...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
1. Let X Xn be independent and identically distributed as N(0,9) (here 9 is the variance)...
1. Let X Xn be independent and identically distributed as N(0,9) (here 9 is the variance) (a) What is the distribution of Y-X,? (Verify it using MGF) (b) What is the distribution of Xn? (Again verify it using MGF) (c) Assume 25. What is the probability that an observed value of Xn lies inside the interval [-1.2,1.2 (d) Give a lower bound on the probability that Xn lies inside the interval [-1.2,12 using Chebyshev's inequality. Compare it with part (c)....
2. Let X1, X2,. . , Xn denote independent and identically distributed random variables with variance...
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
Let X and Y be two independent and identically distributed random variables with expected value 1...
Let X and Y be two independent and identically distributed random variables with expected value 1 and variance 2.56. First, find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). Now suppose that X and Y are independent and identically distributed N(1,2.56) random variables. What is P(|X + Y − 2| ≥ 1) exactly? Why is the upper bound first obtained so different from the exact probability obtained?
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables,...
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
Problem 7. Let Xi, X2,..., Xn be i.i.d. (independent and identically distributed) random variables with unknown...
Problem 7. Let Xi, X2,..., Xn be i.i.d. (independent and identically distributed) random variables with unknown mean μ and variance σ2. In order to estimate μ and σ from the data we consider the follwing estimates n 1 Show that both these estimates are unbiased. That is, show that E(A)--μ and
Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables...
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 =...
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed,...
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
1. Let X,.., Xn be independent and identically distributed as N (0,9) (here 9 is the...
1. Let X,.., Xn be independent and identically distributed as N (0,9) (here 9 is the variane (a) What is the distribution of Y-1X,? (Verify it using MGF) (b) What is the distribution of Xn X? (Again verify it using MGF) (c) Assume n -25. What is the probability that an observed value of X lies inside the interval [-1.2,1.2] (d) Give a lower bound on the probability that Xn lies inside the interval1.2,1.2] using Chebyshev's inequality. Compare it with...
Let X1, ..., Xn be i.i.d. [Recall that i.i.d. stands for independent and identically distributed.] Since...
Let X1, ..., Xn be i.i.d. [Recall that i.i.d. stands for independent and identically distributed.] Since X1, ..., Xn all have the same distribution, they have the same expected value and variance. Let E(X1) = µ and V ar(X1) = σ 2 . Find the following in terms of µ and σ 2 . (a) E(X2 1 ). Note this is not µ 2 ! (b) E( Pn i=1 X2 i /n). (c) Now, define W by W = 1...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
1. Let X Xn be independent and identically distributed as N(0,9) (here 9 is the variance) (a) What is the distribution of Y-X,? (Verify it using MGF) (b) What is the distribution of Xn? (Again verify it using MGF) (c) Assume 25. What is the probability that an observed value of Xn lies inside the interval [-1.2,1.2 (d) Give a lower bound on the probability that Xn lies inside the interval [-1.2,12 using Chebyshev's inequality. Compare it with part (c)....
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
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
Problem 7. Let Xi, X2,..., Xn be i.i.d. (independent and identically distributed) random variables with unknown mean μ and variance σ2. In order to estimate μ and σ from the data we consider the follwing estimates n 1 Show that both these estimates are unbiased. That is, show that E(A)--μ and
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 =...
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
1. Let X,.., Xn be independent and identically distributed as N (0,9) (here 9 is the variane (a) What is the distribution of Y-1X,? (Verify it using MGF) (b) What is the distribution of Xn X? (Again verify it using MGF) (c) Assume n -25. What is the probability that an observed value of X lies inside the interval [-1.2,1.2] (d) Give a lower bound on the probability that Xn lies inside the interval1.2,1.2] using Chebyshev's inequality. Compare it with...
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