Statistic question for signals please help. Let Z(t)=X(t)+Y(t), where X(t) and Y(t) are independent random signals. Derive the formulas for: i) uz (t). ii) Rz (t, t+T)
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- continuous random variables Z is known to have an Erlang (2,1.3) type PDF. what is the variance of Z ?Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s^2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix using GLMLet X and Y be jointly distributed random variables. This exercise leads you through a proof of the fact that −1 ≤ ρX,Y ≤ 1. a) Express the quantity V(X − (σX/σY)Y) in terms of σX, σY, and Cov(X, Y). b) Use the fact that V(X − (σX/σY)Y) ≥ 0 and Cov(X, Y) = ρX,YσXσY to show that ρX,Y ≤ 1. c) Repeat parts (a) and (b) using V(X + (σX/σY)Y) to show that ρX,Y ≥ −1.
- U is a uniform (0, 1) random variable and X = −ln(1 − U). Find the PDF, CDF of randomvariable X. Calculate the expected value of X.Suppose X and Y are continuous random variables such that the pdf is f(x,y)=x+y with 0≤x≤1,0≤y≤1. a)Draw a graph that illustrates the domain of this pdf. b)Find the marginal pdfs of X and Y.c)Compute ux,ux,var x,var y, cov(x,y) ,and pd (Rho d)) Determine the equation of the least squares regression line and draw it on your graph.Exercise 4.3 Two measurement methods are used to evaluate the surface smoothness of a paper product. Let X and Y denote the measurements of each of the two methods. Suppose that the joint p.d.f of X and Y is modeled by fxy (x, y) = c, 0 <x <4, 0<y and x-1 < y <x+1 Determine the value of c. Determine the marginal probability distribution of X. Find also the mean and variance of X. Determine the conditional probability distribution of Y given X=2. Find also the conditional mean and conditional variance of Y given X=2 Check if the measurements of the two methods X and Y are independent. * SHOW COMPLETE SOLUTION WITH GIVEN, REQUIRED, AND SOLUTION. SHOW TABLE OF ILLUSTRATIONS IF NECESSARY.
- b) Let Y1, Y2, ..., Yn be a random sample from a population with probability density function in part a). Show that the best test for the hypothesis in part a) rejects Ho if |yi <c i=1 where c solves the probability equation a = P(II1Yis c[0 = 2). c) Let X1,X2, ..., Xn be a random sample from GAMMA(2,ß) distribution, and consider Y = E-,Xi- Show whether or not Y is a pivotal quantity and give its distribution.X is a discrete uniform (1,3) random variable. When X=x, Y is discrete uniform (1,x). (a) Find the correlation of X and Y (b) Find the mean value of X when Y=2.Solve using the moment-generating function technique. Let X1, . . . , Xn be independent random variables, such that Xi ∼ N(µi, σ2) for i = 1, . . . , n. Find the distribution of Y = a1X1 + · · · + anXn.
- Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Cov (X,Y) = E[XY] - E[X] E[Y]Two archers will simultaneously shoot once at a target. Let Xi , i = 1, 2, be marksman i's horizontal error, measured in meters. If the target's position is (a, b), and marksman i's shot ends up at (x, y), then Xi = (x - a). These errors are modeled by X1 ~ N(0, 0.5) and X2 ~ N(0, 1). What is the probability that both shots will be within 1 meter of the target, in the horizontal direction?Exercise 2: Let X be a random density variable f(x) = { e−2θx if x ≥ 0 0 else Find θ so that f is a probability density. Determine the distribution function F of the variable X. Calculate the expectation and variance of X. Let Y be the random variable defined by Y = θX. What is the law of Y?
