Find the invariant probability, Basic Statistics

Problem 1. Let X0;X1;X2. . . be a Markov chain with state space f1; 2g and transition probabilities given as follows:

p11 = 0:3; p12 = 0:7; p21 = 0:5; p22 = 0:5:

Find the invariant probability π of the chain. Find P(X2 = 1;X3 = 2 j X0 = 1).

Problem 2. Let X be a random variable such that

M(s) = a + be2s + ce4s; E[X] = 3; var(X) = 2:

Find a, b and c and the PMF of X.

Problem 3. Let X and Y be independent exponential random variables with a common parameter λ. Find the moment generating function associated with aX + Y , where a is a constant.

Problem 4. Let Y be exponentially distributed with parameter 1, and let Z be uniformly distributed over the interval [0; 1]. Find the PDFs of W = Y - Z and that of X = [Y - Z].

Problem 5. You roll a fair six-sided die, and then you flip a fair coin the number of times shown by the die. Find the expected value of the number of heads obtained.

Problem 6. The random variables X1. . . ;Xn have common mean μ, common variance σ2 and, furthermore, E[XiXj ] = c for every pair of distinct i and j. Derive a formula for the variance of X1 + . . . + Xn, in terms of μ, σ2, c, and n.

Problem 7. Consider n independent tosses of a die. Each toss has probability pi of resulting in i. Let Xi be the number of tosses that result in i. Find the covariance of X1 and X2.

Problem 8. We are given that E[X] = 1, E[Y ] = 2, E[X2] = 5, E[Y 2] = 8, and E[XY ] = 1. Find the linear least squares estimator of Y given X.

Problem 9. Let U and V be independent standard normal random variables, and X = U + V, Y = U - 2V. Find E[X | Y ], and cov(X, Y ).

Problem 10. A security guard has the only key which locks or unlocks the door to ENS. He visits the door once each hour on the hour. When he arrives: If the door is open, he locks it with probability 0.3. If the door is locked, he unlocks it with probability 0.8. After he has been on the job several months, is he more likely to lock the door or to unlock it on a randomly selected visit? With the process in the steady state, Joe arrived at Building 59 two hours ahead of Harry. What is the probability that each of them found the door in the same condition? Given the door was open at the time the security guard was hired, determine the expected value of the number of visits up to and including the one on which he unlocks the door himself for the first time.

Posted Date: 2/22/2013 12:57:41 AM | Location : United States







Related Discussions:- Find the invariant probability, Assignment Help, Ask Question on Find the invariant probability, Get Answer, Expert's Help, Find the invariant probability Discussions

Write discussion on Find the invariant probability
Your posts are moderated
Related Questions
what are the procedure of job order costing

The average cost per night of a hotel room in New York City is $273 (SmartMoney, March 2009). Assume this estimate is based on a sample of 45 hotels and that the sample standard


i need help it says for the tables of values determine appropriate veiwing window, create a scatter plot, write a function rule, and graph your funtion rule over your scatter plot

Can you do my accounting assignment?


Authorized travel period Then period when the visitor is in authorized travel position away from the established stop and established property. Talk about OFFICIAL STATION, OFFIC

Budgeted accounts Accounts that are theme to the misappropriation or sometimes to appropriation and/or allotment process. Refer to APPROPRIATION and portion.

Midrange The midrange is the measure of the center that is the value midway between the highest and the lowest values in the original data set. It is found by adding the highes