4 (30') The continuous type random variables X and Y have joint pdf for 0
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- Please help find the pdf for YQ5) A random process is given by X(t) =P+ Q cos(@t+ 0) where P, Q are independent random variables. The mean and variance of distribution of random variable P are 0 and 25 respectively. The mean and variance of distribution of random variable Q are 0, 36 respectively. The angular frequency o is constant and 0 is independent of X(t). The PDF of 0 is given by f(0) = 1 2n 1/2 <0< 31/2. Prove that the random process X(t) is a wide sense stationary random process.Q6. (a) Let X; (i = 1, ..,n) be independent random varables E(X;) = µ and var(X;) = of (i = 1, .,n). Let a = E-1 a; X; be an unbiased estimator of u where a; are constants. Find out how the a, are to be chosen if you wish to minimize var(@). (b) The bias and the mean squared error of an estimator Ô of a parameter 6 are defined as Bias (Ô) = b(ê ) = E(@) – 0 and m.s.e (@) = E{(@ – 0)?}, respectively. Show that m.s.e (@) var@) +{b(@ )}°.
- 34. Consider the continuous random variable X whose pdf is given by f(r) = (a-r³)Io<<1(1). (a) Find the value of a that makes f(z) a pdf. (b) Find E(X). (c) Find V(X). (d) Find P(X ≤).A random variable Y have a distribution with parameter a > 0 and yo > 0 if its density function is given by: ´ayf f(y) = {ya+1 (0, if y > yo elsewhere ii) ayg (a-2)(a-1)2" Show whether or not that the variance of Y isAsap
- (iv) (v) (vi)Exercise 6. Using the MGF given by Equation (24), deduce the expressions for the mean and variance we found earlier. Exercise 7. Prove the algebraic identity: 2g2 + xt = + ut 25 2g22. 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(b) Let X1, X2, ...,X, be a random sample from the pdf Be z for z 2 0 f(z,0) = 0, otherwise (1) Show that the statistic T(X1, X2, . . , Xn) = X, is complete sufficient for 8. %3D i=1(1) Let X = b(16, -) find E(4-3x) and distribution function. (2) Let X be a random variable with p.d.f. -2 kx 1SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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