Let{N(t),t≥0}beaPoissonprocesswithrateλ.Fori≤nands<t,
(a) findP(N(t)=n|N(s)=i);
(b) findP(N(s)=i|N(t)=n).
Let{N(t),t≥0}beaPoissonprocesswithrateλ.Fori≤nands<t, (a) findP(N(t)=n|N(s)=i); (b) findP(N(s)=i|N(t)=n).
a. Let B be an n x n Orthogonal matrix, that is B^-1 = B^T, and let A be an n x n skew-symmetric matrix. Simplify A(A^2(BA)^-1)^T b. Let A be a square matrix such that A^3 = 0. A is then called a nilpotent matrix. Define another matrix B by the expression B = I - A; Show that B is invertible and that its inverse is I + A + A^2 c. Let B = (-2,0,0 ; 0,0,0...
Let n be a positive integer, and let s and t be integers. Then
the following hold.
I need the prove for (iii)
Lemma 8.1 Let n be a positive integer, and let s and t be integers. Then the following hold. (i) We have s et mod n if and only if n dividest - s. (ii) We have pris + t) = Hn (s) +Mn(t) mod n. (iii) We have Hr(st) = Hn (3) Men(t) mod n. Proof....
f(t) F(S) (s > 0) S (s > 0) n! t" ( no) (s > 0) 5+1 T(a + 1) 1a (a > -1) (s > 0) $4+1 (s > 0) S-a 1. Let f(t) be a function on [0,-). Find the Laplace transform using the definition of the following functions: a. X(t) = 7t2 b. flt) 13t+18 2. Use the table to thexight to find the Laplace transform of the following function. a. f(t)=t-4e2t b. f(t) = (5 +t)2...
Let (N(t), t ≥ 0) be a Poisson process with rate λ > 0. Show that, given N(t) = n, N(s), for s < t, has a Binomial distribution that does not depend on λ, justifying each step carefully. What is E(N(s)|N(t))?
6. Let S(t) denote the price of a security at time t. A popular model for the process S(t): t 0} supposes that the price remains unchanged until a "shock" occurs, at which time the price is multiplied by a random factor. If we let N(t) denote the number of shocks by time t, and let Xi denote the ith multiplicative factor then this model supposes that N (t) "x II (0)s ()s
6. Let S(t) denote the price of...
Let N(t), t 2 0} be a Poisson process with rate X. Suppose that, for a fixed t > 0, N (t) Please show that, for 0 < u < t, the number of events that have occurred at or prior to u is binomial with parameters (n, u/t). That is, n. That is, we are given that n events have occurred by time t C) EY'C)" n-i u P(N(u) iN (t)= n) - for 0in
Let N(t), t 2...
Let X,, X,,... be independent and identically distributed (iid) with E X]< co. Let So 0, S,X, n 2 1 The process (S., n 0 is called a random walk process. ΣΧ be a random walk and let λ, i > 0, denote the probability 7.13. Let S," that a ladder height equals i-that is, λ,-Pfirst positive value of S" equals i]. (a) Show that if q, then λ¡ satisfies (b) If P(X = j)-%, j =-2,-1, 0, 1, 2,...
0 #UF23 Let s," Show that, for n 22, s (a) ," >S+, (b) Deduce that Spm>S,+ (c) Hence show that the sequence S.) is divergent.
0 #UF23 Let s," Show that, for n 22, s (a) ," >S+, (b) Deduce that Spm>S,+ (c) Hence show that the sequence S.) is divergent.
Let the Brownian motion (b(t)) start at X0 (constant) B(0)=X0, => B(t) ~ N(X0,t) why?
Problem 2. Let 1 1-i 1+i 0 T= (a) Verify that T is hermitian. (b) Find its eigenvalues and corresponding (normalized) eigenvectors (d) Construct the unitary diagonilizing matrix S and explicitly evaluate STS-1