Does the function P(x) = 1 for x = 0,1,2 Constitute a probability function. Unspecific Criteria...
3. A random variable X has the probability mass function P(x = k) = (a > 0, k =0,1,2...). (1 + a)! Find E[X], Var(X), and the Moment generating function My(t) = E[ex]
If the joint probability distribution of X and Yis given by: fxy)-2xty48,for all x-0,1,2,3 and y-0,1,2 Determine Part a: P(Xs3,Y-1) Part b: P(X+Y-4) Part c: Part d: E(XY)]
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (pij). Define T = min(n > 0 : Xn-0), and vi(n) = P(T > n|X0 = i). Use the first-step analysis to show that vi (72), t"2(n), . . . , UN(n)) = where B is a submatrix of P obtained by deleting the row and column corresponding to the state 0. Hint: First establish a recursive formula v(n )-ΣΝ1pijuj(n-1). Consider a...
A random variable X has a distribution with probability function f(x) = K(nx)2x for x = 0,1,2,...,n where n is a positive integer. a. Find the constant k. b. Find the expected value M(S) = E(esX) as a function of the real numbers s. Compare the values of the derivative of this function M'(0) at 0 and the expected value of a random variable having the probability function above. c. What distribution has probability function f(x)? Let X1, X2 be independent random variables both...
Consider the geometric random variable X with probability mass function P(X =x)=(1 p)x 1p, x=1,2,3,.... For t <- l o g ( 1 - p ) , c o m p u t e E [ e t X ] .
1. An application in probability (a) A function p(q) is a probability measure if p(x) > 0VT E R and (r) dx = 1. We first show that p(x):= vino exp(-) is a probability measure. (1) Compute dr. (ii) Show that were dr = 1. (ii) (1pt) Conclude that pr(I) is a probability measure. (b) A random variable x(): R + R is an integrable function that assigns a numerical value, X(I), to the outcome of an experiment, I, with...
Let Xn be a Markov chain with state space {0,1,2}, the initial probability vector and one step transition matrix a. Compute. b. Compute. 3. Let X be a Markov chain with state space {0,1,2}, the initial probability vector - and one step transition matrix pt 0 Compute P-1, X, = 0, x, - 2), P(X, = 0) b. Compute P( -1| X, = 2), P(X, = 0 | X, = 1) _ a. 3. Let X be a Markov chain...
Recall that the PMF of the Poisson distribution is P(X-x)- e-1t(at?x -0,1,2.... x! a) b) c) Determine the probability that no goals are scored in the game. Determine the probability that at least 3 goals are scored in the game. Consider the event that the game is tied, i.e. both teams score equal numbers of goals. Is it true that P(“game is tied") = P("X is even")-Σ000 P(X-2x)? Justify your answer.
Assume that the probability mass function of X is given by P(X = 1) = P(X = 2) = P(X = 3) = 1/3 A random sample of n = 36 is selected from this population. Find the probability that the sample mean is greater than 2.1 but less than 2.5, assuming that the sample mean would be measured to the nearest tenth.
Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint: Determine first the probability distribution of T -X +Y + Z using the moment generating function method. Moment generating function for Poisson random variable is given in earlier lecture notes) Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint:...