Example 3.17 Let X be a discrete random variable with range Rx = {0, 5,5, , 7}, such that Px(0) = Px(÷) = Px(5) = Px() = Px(#) =. Find E[sin(X). %3D %3D %3D %3D %3D
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- 3-3.2 Two rv X and Y are independent, if f(x) = 0.5e-, and f(y) = 0.5el-1 (no range for both), find P(XY > 0). Also, construct the correlation and the covariance matrices. b) If X and Y are independent Gaussian rvs with zero mean for both and the variances of 16 and 36 respectively, find P(XY<0).2 163Example 7: Given that the random process X (t) = 10 cos (100t + 0) where o is a uniformly distributed random variable in the interval (- a , n). Show that the process is correlation-ergodic.
- Part III: Sections 3.1- 3.3 13. Suppose that X is a continuous random variable with pdf given as 43 if 1< <4 255 f(1) = %3D otherwise. (a) For 1Example 3.6. Let X and Y be jointly continuous random variables with joint PDF is given by: 3 fx,x(r, y) = (1 + a°y)[(0,1)(=)[(0,2) (1) Note that fx(x) =÷ (1 + a²) I0,1)(x) fr(y) =- (1+y) I(0,2) (y) Solve for fxjy(æ | y). Remark. It follows immediately from the definition of conditional probability density function that fx,x(x, y) = fxjy(x | y) · fy(y), -o∞ < x < ∞ fx,y (x, y) = fy|x(y | æ) · fx(x), -(x₂-μ)² 4(0²)² condition for deriving the estimator for the variance. Consider True a log L 062 False n 20² + i=1 (2) = 0. This is a first orderQuestion 6 Let X be a binomial random variable with E[X] = 7.2 and Var(X ) = 2.88 Find а) Р{X 3D 4} b) P{X > 12}.Question 3 Let X1, XN be random sample of chi-squared random variables each with 3 degrees of freedom. .... What is the moment generating function of Y= X1 + X2+...+Xn. (1-2t)^(-y/2) O (1-2t)^(-N/2) O (1-2t)^(-3/2) O (1-2t)^(-3N/2)Question 20 103 and o = 5. Determine the following Assume X is a normally distributed continuous variable with u = probability. Round the solution to four decimal places, if necessary. P(z < 88) = Question Help: Video 1 Video 2 D Video 3 Submit Question Type here to searchQuestion 7 a) Let X1, X2, ... be an infinite sequence of independent and identically distributed Bernoulli random variables with parameter p = 1/2. Find n lim P (X; b) Let X - U (0,1) and Y - U(0,1), and X and Y are independent. i) Find E[XY]. ii) Find Var[Xx¥].Example 3.6. Let X and Y be jointly continuous random variables with joint PDF is given by: 3 fx.x(x, y) =(1+ a²y)!(0,1)(x)/(0.2)() Note that fx(x) = (1+ x²) I0,1) (2) fy(y) : (1+ y) I(0,2)(y) Solve for fx|y(x | y). Remark. It follows immediately from the definition of conditional probability density function that fx,y(r,y) = fxjy(x | y) · fy (y), fx.x (x, y) = fy|x(y | x) · fx(x), -00Question 1 Let X be a continuous random variable taking values in [a, b] with c.d.f. Fx which is strictly increasing on [a, b]. (a) Show that the random variable Fx (X) has a uniform distribution on [0, 1]. (b) Let U be a uniform random variable on [0, 1]. What is the distribution of the random variable F¹(U), where F¹ is the inverse of Fx? (c) Suppose that U₁, U2,..., Un are a set of computer-generated pseudo-random numbers (assumed to be drawn from a uniform distribution on [0, 1]). How would you use them to simulate a random sample X₁, X2,..., Xn from the distribution with density f(x) = με μπ x ≥ 0?SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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