A uniformly distributed random variable, X, described below, is input into a system described by the function g(x) forming a new random variable Y. Determine the CDF for Y. Show your work! y=g(x)
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- Find X2 (the probability distribution of the system after two observations) for the distribution vector X0 and the transition matrix T. X0 = 0.9 0.1 , T = 0.1 0.6 0.9 0.4 X2 =Let the stochastic process {Xt} be defined as Zt ; if t is even (Z2t-1 -1)=21/2; if t is uneven, where {Zt} is identically and independently distributed as Zt is N(0, 1). Show that {Xt} is WN(0, 1), but not IID (0,1).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.
- 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.Use the expected value properties to obtain the E[Y] of the following system : y = 3x + 1 , where E[X] =3LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.
- The number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are under study. Suppose that the joint pmf of X and Y is modeled as fxy (x,y)=c(x+y), x=1,2,3 and y=1,2,3. Check if the number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are independent.Pij = P(Xn+1 =j|Xn=i) Xn is a Markov chain Show that for a fixed i, Sum of jPij = 1X1 and X2 are two discrete random variables, while the X1 random variable takes the values x1 = 1, x1 = 2 and x1 = 3, while the X2 random variable takes the values x2 = 10, x2 = 20 and x2 = 30. The combined probability mass function of the random variables X1 and X2 (pX1, X2 (x1, x2)) is given in the table below a) Find the marginal probability mass function (pX1 (X1)) of the random variable X1.b) Find the marginal probability mass function (pX2 (X2)) of the random variable X2.c) Find the expected value of the random variable X1.d) Find the expected value of the random variable X2.e) Find the variance of the random variable X1.f) Find the variance of the random variable X2.g) pX1 | X2 (x1 | x2 = 10) Find the mass function of the given conditional probability.h) pX2 | X1 (x2 | x1 = 2) Find the mass function of the given conditional probability.i) Are the random variables X1 and X2 independent? Show it. The combined probability mass function of the random variables X1 and X2 is below
- Let A(t) be a n × n matrix whose coefficients are continuous functions of t on the interval (α, β). Let t0 ∈ (α, β) and asssume that y(t) is a solution of the initial value problem x' = A(t)x, x(t0) = 0. Show that y(t) = 0 for all t ∈ (α, β).Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?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.