Let theta ~ Unif([0, 2pi]) and consider X = cos(theta) and Y = sin(theta) .
(i) Determine the correlation coefficient between X and Y .
(ii) Prove that X and Y are not independent.
Let theta ~ Unif([0, 2pi]) and consider X = cos(theta) and Y = sin(theta) . (i)...
= Let cos(6) sin(0) B - sin() cos() and 0 << 27 (i) Calculate the eigenvalues of B. Hence prove that the modulus of the eigenvalues is equal to one. (ii) Calculate the eigenvectors of B.
Find the correlation between the two given signals. x(t)=sin(2pi*t) pi(t-0.5) y(t)=cos(2pi*t) pi(t-0.5) pi(t) is the unit rectangular pulse Please write legibly and explain
1. Consider the uniform distribution X defined over the interval [0, 2pi]. Now let Y = sin(X) (a) Calculate the CDF FY(y) of Y. (b) Calculate the PDF f(y) of Y. In particular, in what interval [a, b] is Y defined? (this mean f(y) = 0 for y < a and for y > b). (c) Verify that f(y) is a PDF.
Let X have the pdf defined for 0<x<2. Let Y~Unif(0,1). Suppose X and Y are independent. Find the distribution of X-Y. fx() =
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy. dz= Show that dz is an exact differential and determine the corresponding function f(x,y) Hence solve the differential equation = z sin( Cos( y) 2 x cos( y) dy 10] (b) Find the solution of the differential equation d2y dy 2 y e dx dæ2 initial conditions th that satisfi 1 (0) [15] and y(0) 0 4. (a Let (sin( x cos( ) dr...
= Consider the vector field F(x, y) (cos y + y cos x)i + (sin x – xsin y)j. Show whether the function f(x,y) = x COS Y – y sin x is a potential function for the vector field, F.
let Theta be an angle in quadrant IV with cos(theta)=3/5 find sin(2theta), cos(2theta) and tan(theta/2)
Suppose that Y=cos(X), where X is uniformly distributed over the interval [0, 2Pi]. Determine the pdf of the random variable Y.
Show how [sin((x+0.5)theta) - sin((x-0.5)theta)] / [2sin(theta/2)] = cos(x theta) Please show all steps clearly. Please do not overlook the theta in two places in the numerator.
Let X ~ N(0, 1), and let Z ~ Unif{-1, 1} (i.e. P(Z = -1) = P(Z = 1) = 1/2) be independent of X. Let Y = ZX. What is the distribution of Y? Show that X and Y are uncorrelated. Are X and Y independent?