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
SOLVE THE FOLLOWING PROBLEM IN SPSS. A college is conducting a study of its students' expectations of employment upon graduation. Students are sampled by class and major area of s

You are working for the management of ESK Jewellery as a marketing consultant. The main business of ESK Jewellery is the retailing of diamond jewellery. They opened their first sho

S&V Industries manufactures book cases. Different sizes of cases are kept in inventory. The company has 80 man-hours and 36 pounds of wood available each day. 2 pounds of wood are

What is the net book value of a noncurrent asset? The net publication value of a noncurrent resource is the net quantity revealed on the stability piece for a long-term resourc

formula for calculating paasches quantity index

if a series of three observations has a mean andd median of 4 and mean absolute deviation of 4/3. what is the series

In November and December 2010, Lane Co., a newly organized magazine publisher, received $90,000 for 1,000 three-year subscriptions at $30 per year, starting with the January 2011 i

how do you interpret the correlation on the basis of probable error

What is the relationship between the normal curve and the standard deviation? Name and explain three special features of a normal distribution