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PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic differential equation dS ơSdX+1Sdt. what stochastic differential equation does the stochastic process (a) Y 25, (b) Y = S (c) Y-es, (d) YeT-/S follow? In each cases express the coefficients of dX and dt in terms of Y rather than S. Use Ito's lemma
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic...
11. Simplify completely a. (3 pts) 85 b. (3 pts) 16 DS c. (3 pts) 252 12. (3 pts) Use properties of exponents to simplify completely the expression below. 13. Simplify completely. a. (3 pts) 20y* b. (3 pts) 18
(6 pts) Let f(x) = (x2 + 3x + 1)e-x. (a) (1 pt) Find f'(2) (b) (3 pts) Solve for the intervals of increase and decrease. Show your work. (c) (2 pts) Find any local maxima or minima, and where they occur.
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all a,b>0 Moreover, assume that f(0)1 (a) Prove that there exists λ (not necessarily positive) such that f(r) = e-Ar, for all r. Hint Find and solve a proper differential equation. (b) Suppose that X is a continuous random variable, with P(X>ab)-P(>a)P(X> b), for all a, b e (0, oo). Prove that X is exponentially distributed
1. (8 pts.) Suppose T is a linear transformation and -(1)-(C). --- () - 0 Solve the matrix vector equations below. Explain 2. (8 pts.) Below are a matrix A, its inverse, and a vector b. 1983--013... (1) A = A-1 = 1-4 3 2 -5 7 2 0 1) , and b = x) Determine the value of x. Then solve the matrix vector equation Av = b. [-u-5 il 14 s1 3 70 | 71-3-70 12 x 12...
7. (15 pts) Suppose X1, X2, ..., X, is a random sample from an exponential distribution with parameter 2. (Remember f(x;2) = ne-^x is the pdf for the exponential dista.) a) Find the likelihood function, L(X1, X2, Xn). b) Find the log-likelihood function, I = log L. c) Find d //d, set the result = 0 and solve for 2.
just answer e through h
8. (11 pts) Let (Xn) be a sequence in Rº such that VnEN, Xn+1 = A· Xn+ where A = (5/8 5/3) and Xo = (-1) (a) (1 pt) Find X1. (b) (2 pts) Find the corresponding equilibrium point. (c) (1 pt) Determine the two eigenvalues 11 and 12 of A. (d) (1 pt) For each eigenvalue, find an eigenvector. (e) (1 pt) Is the equilibrium point a sink? Justify. (f) (1 pt) Deduce the...
2. (2 pts) Suppose X follows a Gamma distribution with parameters a, B, and the following density function F(t) = f(a)ga Find o and 8 so that E(X) = Var(X) = 1. 3. (2 pts) Find the median for the random variable, X. in #2.
1. Add, subtract, or multiply as indicated. Simplify completely. (4 pts) (x - 4) - (x2 – 3x + 4) a. b. (4 pts) (x + 5)(x2 + 2x) C. (4 pts) (x - 2)2 2. Solve each equation using any method. (4 pts) 2x2 + 4x - 30 = 0 a. b. (4 pts) (x - 3)2 = 16 C. (4 pts) x² - 4x = 6 d. (4 pts) x2 = 81 e. (4 pts) 72x = 8x2...
Suppose X is a random variable such that E(X) and E(X2 ) both exist, and are finite. Consider the function f(c) of a real number c given by f(c) = E[(X ? c)2 ]. (a) (2 pts.) Find this function f(c) when X ? Bin(3, 1/2). Among the ’zoo’ of functions that you know about, what kind of function is it? (b) (8 pts.) Find the value of c which MINIMIZES the function f(c). Hint: expand out the (X ?...