A random process X (t) is defined by X (t) = 2. cos (2 nt + Y), where Y is a discrete random variable with P (Y = 0 ) = 1 and P (Y = T/2) = Find E [X (1)] and 2 Ryx (0, 1).
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- 1 Suppose that X is a stochastic process with dynamics dXt = µdt +σdWt , where W is a P-Brownian motion. The drift µ and the volatility σ are both constants. Find if there is a measure Q such that the drift of process X under Q is η(∈ R) instead of µ.If X is exponentially distributed with parameter λ and Y is uniformly distributed on the interval [a, b], what is the moment generating function of X + 2Y ?Consider a random process X(t) defined by X(t) = U cos t + (V + 1) sin t, −∞ < t < ∞where U and V are independent random variables for which E(U) = E(V) = 0 E(U2) = E(V2) = 1(a) Find the autocovariance function KX(t, s) of X(t).(b) Is X(t) WSS?
- Consider the geometric Brownian motion with σ = 1: dS = μSdt + SdX, and consider the function F(S) = A + BSα. Find any necessary conditions on A, B, and α such that the function F(S) follows a stochastic process with no drift.A rectangular plate with insulated surface is 10 cm. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u(x,0) = 8 sin(px/ 10) when 0 <x <10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0o C, find the steady state temperature distribution u(x,y).Assume an asset price S_t follows the geometric Brownian motion, dS_t = µS_tdt + σS_dW_t, where µ and σ are constants and r is the risk-free rate. 1. Using the Ito’s Lemma find the stochastic differential equation satisfied by the process Xt = S_t^n , where n is a constant. 2. Compute E[X_t] and Var[X_t]. 3. Using the Ito’s Lemma find the stochastic differential equation satisfied by the process Y_t = S_tert
- Prove the following property of the compound Poisson process:1. E(xt) = λ t E(Y).Consider the autonomous DE . dx/dt = x^2 - 8x + 7 (a) Its stable critical point is x= unstable critical point is x= (b) Applying five iterations of the Forward Euler Method to the above DE, using a step size h(0.1) and initial condition x(0)=3, yields: x(0.1)approximately x1 = x(0.2)approximately x2 = x(0.3)approximately x3 = x(0.4)approximately x4= x(0.5)approximately x5 =Let X1 ... Xn i.i.d random variables with Xi ~ U(0,1). Find the pdf of Q = X1, X2, ... ,Xn. Note that first that -log(Xi) follows exponential distribuition.
- B) Let dP/dt =.5P - 50. Find the equilibrium solution for P. Furthermore, determine whether P is intially increasing faster if the initial population is 120 or 200.Consider a 6-meter metal bar with a uniform initial temperature (across the bar) of 35°C . Suppose it is in thermal contact with an external source of heat given by h(x)= 3−x, 0 ≤ x ≤ 6. So the temperature u(x,t) had the ut=uxx+h(x) permission. Suppose further that the temperature of the ends are kept constant, being at x=0 of 5°C , while at x=6 of 30°C . Under such conditions: Find the steady-state temperature distribution of the bar and the boundary value problem that determines the transient distribution. (no need to solve the problem).f X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + ···+ Xn is a sufficient estimator of the parameter θ.