A series xt generated by the moving average process as: xt = µ + εt + θ1 εt−1, where εt are independently identically distributed random variables with E(εt) = 0, and V ar(εt) = σ2 . (a) Calculate the unconditional mean and the unconditional variance of xt (b) What is meant by saying that a process like xt is invertible? What condition would assure that xt is invertible? If θ = 0.75, does xt satisfy the invertibility condition? (c) What shapes of the ACF and PACF functions do you expect for xt ? Derive the first 4 autocorrelations for this process (τ1 up to τ4). (d) Write the equations for the 1, 2, 3 and 4 step ahead forecasts for xt .

Q: axa-1 f(y|a, 0) 0 0,0 > 0. ga (a) Show that the expected value and variance of Y are a E(Y) = ( 0…

A: Y is a random variable having the following pdf- f(y)=αyα-1θα,0<y<θ, α>0, θ>00,otherwise

Q: Question 1 Let X₁, X, be an i.i.d. sample from the distribution with probability mass func- tion…

A:

Q: 5. For a time series data set with 100 observations. Ŷ2 = 1.76440, Ŷs = 0.85559 and r2 = 0.62805.…

A: Formula for autocorrelation function is given by: r(h) = γ(h) / γ(0) autocorrelation function.…

Q: Question 5 Let X1 and X2 be random sample of normal distribution with mean 1 and 2 and variance 3…

A: Solution

Q: Let x1, X2. ... , Xn d N(4, 62), where HE R and o> 0 are unknown parameters. Consider the sample…

A: Given Xi follows normal distribution and given the formula of sample variance- s2=1n-1 ∑i=1nxi-x¯n2…

Q: (a) A random variable has px(x) = x/15 for x = 1, 2, 3, 4, 5 and 0 otherwise. Find the mean and…

A:

Q: Suppose a time series of returns r, follows the following recursive formula: = (1 — αL)(1 — BL³) at…

A: The question presents a recursive formula for a time series of returns, denoted by rt, which…

Q: Question 13. Let X1 ~ N(1,1) and X2 ~ N(2,4) be two normally distributed and statisti- cally…

A: Let's analyze each statement one by one:(a) P(X1 > 1) = P(X2 (b) Cov(X1, X2) = 0: Since X1 and X2…

Q: Question 5. If the random variable Y has the mgf M (t) = } + et+e, -o <t < o0, then P(-0.2 <Y S1.5)…

A:

Q: Let X1, ..., Xn be iid N (µ, o) and suppose we are interested in estimating t(µ, 0) = o². The…

A: Since there are multiple questions we will answer the first one. Introduction - Normal distribution…

Q: 2. Let U₁, U₂,... be IID Uniform(0, 1) random variables. Let M₁ = I₁U; be the product of the first n…

A:

Q: X is an exponential random variable with a mean of 5 and Y is a Rayleigh variable of second moment…

A:

Q: If the moment-generating function of X is M(t) = {e¹ + find the mean, variance, and pmf of X. 2t

A:

Q: Question is attached.

A: Given thatThe moment generating function of a random variable isTo find mean and variance:

Q: Let X and Y be zero-mean independent Gaussian random variables with common variance o². If Z = √X² +…

A:

Q: (i) Find E(X) and var(X). (ii) Using Σ(X; – X)2 = Σ{(X; – μ) + (μ – Χ)}2 show that n n Σ(x; − x)2 =…

A:

Q: A time series {yt} follows an MA(2) model: Yt = 2 + Ut +0.54t-1 + 0.4ut-2. Assume that ut is a white…

A: Here it is given that a time series yt follows an MA(2) model :…

Q: Let X1, .., XN be random sample of chi-squared random variables each with 3 degrees of freedom. What…

A: We have given that the X1, X2 , . . . .  . XN be random sample of Chi-Squared random variables each…

Q: 1. Aggregate risk S comes from compound Poisson random variable with N- P(200) and individual claims…

A:

Q: (5) If X is a normal N(0, 1) random variable then E(X) is 1 -1 N/A (Select One) (6) Find the…

A: (5) If X is a normal N0,1 random variable. That is, μ=0,σ2=1. The expected value of X is, μ=EX=0…

Q: Consider the following MA(2) process Xt = Zt + θ1Zt−1 + 1/ 8 Zt−2, where θ1 does not equal to 0 is a…

A: To calculate the variance of the sample mean for the MA(2) process with the given value of we'll…