(2)(10pts) A ship has primary navigation system X and backup system Y. X is the event...
The joint density of random variables X and Y is given to be f(x,y) =xy^2 for 0≤x≤y≤1 and is 0 elsewhere. (a) Compute the marginal densities for X and for Y respectively. (b) Compute the expected valueE(XY). (c) Define a new random variable W=Y/X. Compute the probability P(W > t) for anyt >1. Also find the probability P(W <1/2) ?
8) CHALLENGE QUESTION A probability experiment has exactly four mutually exclusive outcomes: W,X,Y & Z. The following is true: • P(W) = 4P(X) • P(X) = P(Y) - P(Z) • P(W) = P(X) + P(Y) + 5P(Z) Find the four probabilities P(W), P(X), P(Y), P(Z)
(45) Two random variables X and Y have the joint probability density | 2, 0sxs1 and 0 s ys1 and x + y21 fxY (x, y) = 0, elsewhere Answer each of these independent questions about X, Y, carefully indicating the domain of all functions where needed. Parts a). - i). are 5 points each. a). Find E(Z), where Z is a new random variable defined by Z = XY b). A is the event {X >0.75}. Find P(A). c)....
dx/dt = 4x -x^2 -2xy dy/dt = -y+0.5 xy a) find equilibrium points b) find Jacobian matrix for above system c) find Jacobian matrix at eq. point (0,0) d) draw phase portrait near (0,0) from © e) show at eq. point (4,0) the Jacobian matrix is -4 -8 0 1 f) draw phase portrait near (4,0) from (d) g) at eq. point (2,1) the Jacobian matrix is -2 -4 0.5 0 h) draw phase portrait near (2,1) from (f) i)...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutions and determine their type (e.g., spiral source, saddle) Hint: you should find three equilibria. b) For each of the equilibria you found in part (a), draw a phase portrait showing the behaviour of solutions near that equilibrium. -2 (c) Find the nullclines for the system and sketch them on the answer sheet provided. Show the direction of the vector field...
1. A LTI system has the frequency response function 0, all other o Compute the output y(t) resulting from the in put x(t) given by (a) x(t) -2-5cos(3t)+10sin(6t-jx/3)+4cos(12t-x/4) (b) x(t) = 1 + Σ- cos(2kt ) k-l (c) x(t) is the periodic pulse train signal shown below (repeats beyond the graph) 0.5 0.5 5 t (second) Hint: Refer to lecture 10 note. For (c), find the Fourier series coefficients of x(t) first.
1. A LTI system has the frequency response...