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Consider a random process X(t) defined by 4 X(t) = Ucost + (V + 1)sint, where U and V are independent random variables for which E(U) = E(V) = 0; E (U²) = E (V²) = 1 a) Find the auto
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- Let m(t) be the moment generating function of a random variable X. Show that the random variable W = 10X is m(10t). What is the moment generating function of Z = X-5 in terms of m(t)?Consider a random process which is given by Y(t) = t - Z where Z is a random variable with mean 1.2 and second moment 2.5. The autocovariance of the random process X(t) isConsider the log likelihood function L(x | 0) where 0 = (01,02)' is a vector of parameters . Let 0* be the value of 0 that makes the gradient of L(x | 0) with respect to 0 equal to vector 0. The hessian matrix evaluated at V3 The log likelihood at 0* is then 1 2 critical 0* is given by H = V3 (a) a local maximum (b) a local minimum (c) a saddle point (d) 0
- Consider a random process X(t) = A cos o t where a' is a constant and A is a random variable uniformly distributed over (0, 1). Find the auto correlation and covariance of X(t).2. Let X and Y be jointly continuous random variables with joint PDF x + cy2 0, OSXS1,0Sys1 elsewhere a) Find the constant c Find the marginal PDF's fy(x) and fy(y) c) Find P(OSXS1/2,0SYS1/2) b)The joint PDF of a two dimensional random variable (X, Y) is given by x? f(x, y) = xy +– 01) (ii) P(y 1 /y 1) (v) P(x(vi) Given below is a joint pdf of X₁ and X2₂ which are independent exponential random variables with parameter X. f(x1, x2) Find the joint probability density function (pdf) of Y₁ = X₁ − X₂ and Y₂ = X₁ + X₂ using the transformation method. =e-21-x2 0 < x1 <∞0,0 < x₂ <∞Q-6 The j.p.d.f of two continuous random variable, x and y are given by f(x,y)= x² + (xy/3), Osx<1, OSys2 Calculate a) Var(x) b) Var(y) c) Cov(x,y) d) Correlation coefficient of X and Y7. Suppose that the random variable X have the pdf f (x) = e-a*/2, x > 0 and 0 otherwise. 2T (a) Find E(X) and Var(X). (b) Find the transformation g(X)= Y and values of a and B such that Y ~ I(a, B).Example 7: Given that the random process X (t) = 10 cos (100t + 0) where 1s a uniformly distributed random variable in the interval (-1 , T). Show that the process is correlation-ergodic. %3DA Random process X(t) is applied to a network with impulse response h(t) = u(t) exp (-bt) where b > 0 is a constant. The cross- correlation of X(t) with the output Y(t) is known to have the same form, Ryy(t) = u(t)t exp(-bt) (i) Find the auto-correlation of Y(t).12. Let the random variable X and Y have joint pdf 4 f(x,y) = (x² + 3y²), 0SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON