Find the steady-state probability vector for the stochastic matrix P. [0.2 0.1] P = 0.8 0.9
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- Find X2 (the probability distribution of the system after two observations) for the distribution vector X0 and the transition matrix T. X0 = 0.25 0.40 0.35 , T = 0.1 0.1 0.2 0.8 0.7 0.4 0.1 0.2 0.4 X2 =X is a discrete random variable and takes the values 0,1 and 2 with probabilities of 1/6, 1/3 and 1/2, respectively. What is the moment generator function M(t) of X?A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability .3, and his second will lead independently to a sale with probability .6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard model, which costs $500. Determine the proba-bility mass function of X, the total dollar value of all sales.
- Find the stable distribution for the regular stochastic matrix. 0.5 0.2 0.3 0.4 0.7 0.7 0.1 0.1 0 Find the stable distribution. (Type integers or decimals rounded to three decimal places as need: We assume that the stochastic process for a stock price is an Arithmetic Brownian motion, with a drift of 53% and, diffusion of 33%. Find the probability that the stock price will be between 0.78 and 1.25 in 4 years. (A) 0.06 (B) 0.04 (C) 0.08 (D) 0.05 (E) 0.07Let A be an n × n stochastic matrix and let e be the vector in Rn whose entries are all equal to 1. Show that e is an eigenvector of AT . Explain why a stochastic matrix must have λ = 1 as an eigenvalue.
- Approximate the stationary matrix S for the transition matrix P by computing powers of the transition matrix P.P is the transition matrix for a Markov chain with two states. X0 is the initial state vector for the population. Find x1 & x2, and find the steady state vector.Find E(R) and V (R) for a random variable R whose moment-generating function ismR(t) = e2t(1-3t2)-1
- Let X be a Gaussian random variable (0,1). Let M = ln(5*X) be a derived random variable. What is E[M]?Let the stochastic process {Xt} be defined as Zt ; if t is even (Z2t-1 -1)=21/2; if t is uneven, where {Zt} is identically and independently distributed as Zt is N(0, 1). Show that {Xt} is WN(0, 1), but not IID (0,1).(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.