x(t) == ()-1 h(t)=pt .u(t) y(t) Z(t) is a sequence of Poisson impulses. X is the expected number of impulses per Find a unit time E[Ÿ(t)] b) Ryy (T).. c) Syy (w)
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![x(t) == ()-1
h(t)=pt
.u(t)
y(t)
Z(t) is a sequence of Poisson impulses.
X is the expected number of impulses per
Find a
unit time
E[Ÿ(t)]
b) Ryy (T)..
c)
Syy
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- Show that the random process X(t) =cos(2π fot + θ) Where θ is an random variable uniformly distributed in the range {0, π/2, π, π/3} is a wide sense stationary process .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 200.1 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.
- 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.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.If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)
