Question

Xt = 2 + Wt - 0.5Wt-1, t= 1,2, 3 ....

where Wt ~ i.i.d N(0,9)

Compute E(Xt) and var(Xt)

Find correlation of Xt and Wt-1

Answer #1

Please take care of one thing here. Generally when we say X follows normal distribution , we read it as X~N(mu, sigma^2)

Here I have taken 9 as sigma^2 for Wt

Consider the series xt = −.8xt−2 + wt and yt = 2cos(2πt) + wt,
where wt ∼ iid N(0,1).
Determine each of the following functions:
(b) the autocovariance functions γx(s, t) and γy(s, t)
(c) the autocorrelation functions ρx(s, t) and ρy(s, t)

Calculate the spectral density of model Xt =
0.5Xt−1+wt+0.5wt−1 where wt is the white noise with variance 1.

Using R statistical software
(a) Generate n = 100 observations from following time series
model xt =2cos(2πt/4)+ωt, where ωt is i.i.d N(0,1). (b) Apply the
moving average to xt and get time series yt, such that yt = (xt +
xt−1 + xt−2 + xt−3)/4. [hint: Use filter(x,rep(1/4,4)),sides=1] (c)
Plot xt as a line and superimpose yt as a dashed line. (d) Describe
the relationship between xt and yt.

X1, … Xn are i.i.d. random variables, and E(Xi ) = 3β, Var(Xi )
= 3β^2 , i = 1 … n, β > 0. Two estimators of β are defined as β̂
1 = (X̅ /3) β̂ 2 = (n /3n+1 ) X̅
Show that MSE(β̂ 2) < MSE(β̂ 1) for a sample size of n =
3.

a) Let Xi for i = 1,2,...n be random variables with E[Xi] = μi
(not necessarily independent). Show that E[∑ni
=1 Xi] = [∑ni =1 μi]. Show from
Definition
b) Suppose that random variables Yi for i = 1, 2,...,n are
independent and identically distributed withE[Yi] =γ(gamma) and
Var[Yi] = σ2, Use part (a) to show that E[Ybar]
=γ(gamma).
(c) Suppose that random variables Yi for i = 1, 2,...,n are
independent and identically distributed with E[Yi] =γ(gamma) and
Var[Yi]...

g(s,t)=f(x(s,t),y(s,t)) where f(x,y)=x^2-xy^3,
x(3,4)=2, y(3,4)=-2, xs(3,4)=4 xt(3,4)=-1, ys(3,4)=10, and
yt(3,4)=-100.
Calculate gs(3,4)

Consider the following difference equation for populations
called the nonlinear logistic model
?t+1=?xt(1−xt)
(a) Find the equilibria. Here ?>0 is an unknown parameter
(i.e., is a constant).
(b) Determine the values of r for which each equilibrium is
stable.

Q2)a) Find the trigonometric Fourier series coefficients of the
following signal
xt=3+4Sin3(10t)
b) Draw the signal defined below and find the trigonometric
Fourier series representation of the signal
xt=n=-∞∞3(-1)nδ(t-4n)

Consider the AR(2) process given by Xt – Xt-1 + 0.25Xt-2 = et.
Find the first five ACF's of the process.
Please post with steps!

Determine the interval(s) where r(t)=<
t-1/((t^2) -1), 2e^3t , In(t+5) > is continuous.
And
Both r1(t) = < t2,
t4 > and r2(t)
= < t4, t8 > map out the same half
parabolic graph.
Notice that both r1(1) = < 1, 1
> and r2(1) = < 1, 1
>.
However, r'1(1) = < 2, 4 >
and r'2(1) = < 4, 8
>.
Explain why the difference is logical and quantify the
difference at t=1.
Determine the interval(s) where r(t)=<t −...

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