Two processes {Z(t)} and {Y(t)} are said to be independent if for any time points t(1), t(2), ... t(m) and s(1), s(2),...s(n) the random variables {Z(t), Z(t2),...Z(tm)} are independent of the random variables {Y(s1), Y(s2),...Y(sn)}. Show that if {Z(t)} and {Y(t)} are independent stationaryprocesses, then W(t) = Z(t) + Y(t) is

Question 1
Suppose that {Y_t} is stationary with autocovariance function ?_k.
a) Show that W_t=?Y_t=Y_t-Y_(t-1) is stationary by finding the mean and autocovariance function for {W_t} .
b) Show that U_t=?^2 Y_t=?[Y_t-Y_(t-1) ]= Y_t-2Y_(t-1)+Y_(t-2) is stationary. (you need not find the mean and autocovariance function for

Assume X(t) is a wide-sensestationary random process with autocorrelation function RX(τ), and that Y(t) is a random process defined by Y(t)= -X(t-t0) where t0 is a constant. The autocorrelation function RYY(t1,t2) of Y(t) is given by what?

Let X(t) be a zero-mean wide-sense stationary Gaussian white noise process with autocorrelation function RXX(τ) = δ(τ). Suppose that X(t) is the input to a linear time-invariant system with an impulse response h(t) = 1[0,T](t) where T > 0. Let Y(t) be the output of the system, and assume that the input has been applied to the

A surface is described by the multivariable function f(x,y) where:
f(x,y) = x^3 + y^3 + 9(x^2 + y^2) + 12xy
a) Show that the four stationary points of this function are located at:
(x1, y1) = (0, 0)
(x2, y2) = (-10, -10)
(x3, y3) = (-4, 2)
(x4, y4) = (2, -4)

Suppose {e_t} and {E_t} are two independent white noise processes with variance sigma^2 and sigma^2_e respectively.
(a) Show that {v_t = e_tE_(t-1)} is also a white noise and calculate its variance.
(b) Show that y_t = v_t - 0.5_(vt-1) + 3e_(t-1) is stationary.

What types of stationary phases used for column chromatography and HPLC? How these staionary phase work and what types of compounds can be separated using this? What are the difference in the stationary phases used for column chromatography and HPLC? Why are they difference and why do not use the materials in HPLC for making low

The problem looks at a general cubic polynomial, and calculates the conditions needed for exactly two stationary points to exist. It also finds the inflexion point.
Consider the cubic polynomial (degree 3) given by, y=ax^3+bx^2+cx+d, where a is not equal to 0.
(a) Find the condition on the constants a,b,c so that this f

Using the function f(x)=2x^3+3x^2-36x+7
a) Find the stationary points of this function.
b) i) Applying First Derivative Test, classify left handstationary point in part a.
ii) Applying First Derivative Test, classify right handstationary point in part a.
c) Find the y coordinates of each stationary point on the graph of th