Proble A Consider t: foMarkower chainXk 0,1,2, where Xk+1, given Xk = n, is Binomial(2n,1/2). Show that the function f(...
Prove that P2n(0)= (-1)n ((2n-1)!!/(2n)!!) using the generation function and a binomial expansion. Show that (sqrt(pi)(4n-1)/(2gamma(n+1)gamma(3/2-n))=(-1)n-1((2n-3)!!/(2n-2)!!)(4n-1)/2n
Consider the Markov chain with state space S = {0,1,2,...} and transition probabilities I p, j=i+1 pſi,j) = { q, j=0 10, otherwise where p,q> 0 and p+q = 1.1 This example was discussed in class a few lectures ago; it counts the lengths of runs of heads in a sequence of independent coin tosses. 1) Show that the chain is irreducible.2 2) Find P.(To =n) for n=1,2,...3 What is the name of this distribution? 3) Is the chain recurrent?...
Question 10. Consider the function defined by f(n) = 2n where n is a positive integer. (i) Can this function be computed by a Turing machine? Why or why not? ( ii) Is this function primitive recursive? Why or why not?
(5) Fixm 2 1, an integer, and suppose P~ Uniform([0, 1]) and N ~Binomial(m, P) (a) Determine E(Xk(NP) where χκ (n), k-0, 1, 2, . . . , are defined as follows: 1 if n-k 0 otherwise (b) Determine E(Xk(N)h(N)) for a general function h : R R (c) Determine E(PIN) Warning: E(PN) is not N/m as you might be tempted to guess. Hint: Use the law of total probability together with the following result which you showed (in greater...
(Sheldon Ross) Consider a process {X,, п : 0, 1, . ( 1, 2, 31, suppose ..1, which takes on the values aij n even, Pj nodd, where j-1 for i = 1, 2, 3. Is {X, | n > 0} a Markov chain ? If not, show how by enlarging the state space, we may transform it into a Markov chain (Sheldon Ross) Consider a process {X,, п : 0, 1, . ( 1, 2, 31, suppose ..1, which...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
Consider a process {Xn, n = 0,1, ... }, which takes on the values 0,1, or 2. Suppose P{Xn+1 = ||Xn = i, Xn-1 = in-1,..., X0 = i +0} = when n is even when n is odd where - P = - Phl = 1, i = 0,1,2. Is {Xn, n = 0,1,... } a time-homogeneous Markov chain? If not, then show how, by enlarging the state space, we may transform it into a time- homogeneous Markov chain.
9. Using the Binomial Theorem, show that Σk㈡-n 2n-1
Consider the function Let where f(t) is differentiable for all t ∈ R. Show that z satisfies the partial differential equation (x2 − y2 ) ∂z/∂x + xy ∂z/∂y = xyz for all (x, y) ∈ R2 \ { (t, 0)|t ∈ R }.
(1 point) Consider the function f(x) = f* cos(t) – 1 dt. t2 Which of the following is the Taylor Series for f(x) centred at x = 0? w A. (-1)" (2n – 1)(2n)! -x2n- +C. n=0 (-1)"(2n – 2) 2n–3. B. (2n)! n=1 c. Σ (-1)" (2n + 1)! -x2n-2 n=1 D. Š (-1)" -X2n-1 (2n – 1)(2n)! n=1