3. (10 pts.) X is a Gaussian random variable with E{X} = 2 and Var(X) = 16. Let Y = 3X +1. Determine the probability: Pr(Y > 2)
1 xe Let f(x)={? x 8. Prove that f(x) continuous only at +1. Let f(x)= $3.x xs! x >1 Using the definition prove lim f(x)=1 and lim f (x) = 3 x>17 11°
Prove that there are no natural number solutions to the equation where x, y ≥ 2 ... (See Picture Below) Prove that there are no natural number solutions to the equation where X, Y > 2. x2 - y2 = 1.
4. (20 points) Use z-transform to solve the difference equation y(k) -1.5y(k-1) + 0.56y(k-2) = x(k) for k> 0 with initial conditions y(-1) = 3, y(-2)=-4, and x(k)= kļu(k).
Solve the equation yu- xui = u, t > 0,x >0 with the initial conditions u(x, 0) =1 + x2 using the method of characteristics. Find the u(x, y). Substitute your found solution u(x, y) in the equation and verify that it satisfies the equation. solution explicitly in the form u =
3) Using the Method of Variation of Parameter, solve the following linear differential equation y' (1/t) y 3cos (2t), t > 0, and show that y (t) 2 for large t
Solve the initial value problem ry' + xy = 1, > 0 y(1) = 2.
Consider the linear system of first order differential equations x' = Ax, where x = x(t), t > 0, and A has the eigenvalues and eigenvectors below. Sketch the phase portrait. Please label your axes. 11 = 5, V1 = 12 = 2, V2 = ()
Solve y'' +9y = $(t – 6), y(0) = y'(0) = 0 g(t) = for t < 6 for t > 6
PROBLEM 1 Let the joint pdf of (X,Y) be f(x, y)= xe", 0<y<<< a. Compute P(X>Y). b. What is the conditional distribution of X given Y=y? Are X and Y independent? c. Find E(X|Y = y). d. Calculate cov(X,Y).