"k)-T, E(X"k+1)-0, k = 0.1, m.g.f. of X and also its ch.f. Then deduce the distribution...
4. Let X1, X, be two r.v.'s with m.g.f. given by t +t 9 12 ' Calculate E(K), σ(K) and C(X1 , X2), provided they are finite.
CALCULATE C(X1, X2) 4. Let X,,X, be two r.v.'s with m.g.f. given by M(t,,ty)=[} (+2 +1) ++ (e? +e?)]”, tų, tz € R.
4. Let X1, X, be two r.v.'s with m.g.f. given by M(t1, tz)=[] (en+2 +1)+] (e? +e?)]”, t1, tz € R. Calculate E(X1), oʻ(X,) and C(X1, X2), provided they are finite.
Let X and Y have the following joint distribution X/Y 0 1 0 0.4 0.1 1 0.1 0.1 2 0.1 0.2 a) Find Cov(4+2X, 3-2Y) b) Let Z = 3X-2Y+2 Find E[Z] and σ 2Z c) Calculate the correlation coefficient between X and Y. What does this suggest about the relationship between X and Y? d) Show that for two nonzero constants a and b Cov(X+a, Y+b) = Cov(X,Y)
1. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. function Let β > 0, δ > O. Consider the probability density fx(x) = βδΧ6-1 e-Px®, x>0, zero otherwise. Find the probability distribution of W-X" b) Determine the probability distribution of W by finding the p.d.f. of W, fw(w), using the change-of-variable technique D) Find the p.d.f. of W, w)-x(w ii What is the name of the probability distribution of W? What are its parameters?...
Problem 0.1 Let X be the number of people who enter a bank by time t>0. Suppose ke-t k! for k 0,1,2,., and for t>s > 0, and k-r=0,1,2, . . . . (a) Find Pr(X2 = k | X,-1) for k = 0, 1, 2, . . . . (b) Find E[X2 X1-1 Useful information: Don't eat yellow snow, and et-L=0 tk/k! Problem 0.2 Recall the Geometric(p) distribution where Xnumber of flips of a coin until you get a...
Use the m.g.f. found in part e) to verify the values of EX and VarX found in part d) Let Y have density function f(y) a) Identify the distribution of γ kye for y 0 (including all parameters) and then use this information to find the valu b) Use the information from part a) to find the moment-generating function for Y c) Use the m.g.f.found in part b) to find EY = e of k and EY ch02, pl grad......
(8) Prove that dt= 1-t n=1 for x e [-a, a],0< a< 1 and deduce from there a power series expansion for -In(1-x) (8) Prove that dt= 1-t n=1 for x e [-a, a],0
please give detail solution. Let X be an r.v. with uniform distribution on [0, 1]. Show that X 2 ∼ Beta(1,1). Let X be an r.v. with uniform distribution on [0, 1]. Show that X2 ~ Beta(3, 1).
1. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. Let β > 0, δ > 0. Consider the probability density function x>0 zero otherwise. Find the probability distribution of W-X c Determine the probability distribution of W by finding the m.g.f. of W, Mw(t) Find the mgf. of w. Mw(t)-E(e, w )-E(e'x®). 1: u-substitution:v5 t X i) Hint -substitution:-»5. Hint', 2: Must have t< for the integral to converge . i What is the...