2. Suppose that X₁, X2, and X3 are mutually independent and identically distributed with moment generating function (m. g. f.) given by M(t). Then, the m. g. f. of X₁ + X2 - 2X3 is given by {M(t)²M(2t)}. True or false, give reason for your answer.
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- 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 ?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.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.
- 4.) Suppose X has moment generating function MX(t) = 0.3e−t + 0.1 + 0.1et + 0.2e2t + 0.3e3t. What is P(X = 2)? What is the P(X = −1)? (To do this problem think about the definition of MX(t) as E(etX) = Σ ext P(X = x)).Find the moment generating function for Y. c=4Determine S[y] has a local maximum or minimum on the stationary path. (by the value of the O(ϵ2) term in Δ)
- What are the equilibrium solutions of the DE dy/dt=4ty−t?If there is more than one, list all of them, separated by commas. If none exist, type DNE.Express the solution of the given initial value problem in terms of a convolution g(t) is an arbitrary function.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.
- Prove the following property of the compound Poisson process:1. E(xt) = λ t E(Y).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.Evaluate the following initial value problem using the Laplace transform and the partial fraction decomposition