X and Y are two independent and identical random variables with moment generating function M(t) You...
5 (10 points) X and Y are independent random variables with common moment generating function M(t) eT. Let W X + Y and Z X - Y. Determine the joint moment generating function, M(ti, t2) of W and Z Find the moment generating function of W and Z, respectively
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
Let X and Y be independent random variables uniformly distributed on the interval [1,2]. What is the moment generating function of X + 2Y? Let X and Y be independent random variables uniformly distributed on the interval [1,2]. What is the moment generating function of X + 2Y?
2. Let Xand Y be random variables with joint moment generating function M(s,t) 0.3+0.1es + 0.4e +0.2 es*t (a) What are E(X) and E(Y)? (b) Find Cov(X,Y) 2. Let Xand Y be random variables with joint moment generating function M(s,t) 0.3+0.1es + 0.4e +0.2 es*t (a) What are E(X) and E(Y)? (b) Find Cov(X,Y)
3. (4 points) The random variables X and Y are independent and have moment generating functions Find Var(X).x (t) =er-2t and Mr(t)=e3t2+tid t a) Find MGF of Z Find Var(Z). Find joint MGF of X and Z, i.e. Mxz(t1,t2) 2X-Y c) d)
Let X and Y be independent random variables, with known moment generang functions Mx(t) and My (t) and Z be such that P(Z = 1) = 1-P(Z 0) = p E (0,1). Compute the moment generating function of the random variable S- ZX (1 - Z)Y. [The distribution of S is called a mirture of the distributions of X and Y.] Your answer can be left in terms of Mx(t) and My (t) Hint: If you don't know how/where to...
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t) exp ( μ1+ σ2t2 ) for -oo < t oo. Show that E X)-ψ(0)-μ and Var(X)-ψ"(0)-[ty(0)12-σ2. 5. Suppose that X1, X2, and X3 are independent random variables such that E[X]0 and ElX 1 for i-12,3. Find the value of E[LX? (2X1 X3)2] 6. Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(X, Y)- 1. Find the value of Var(3X -...
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3) and Y is binomial(n = 3,p = 1/3). Find the moment-generating functions of the three random variables X, Y and X + Y . (You may look up the first two. The third follows from the first two and the behavior of moment-generating functions.) Now use the moment-generating function of X + Y to find the distribution of X + Y .
10. The moment generating function of the random variable X is given by My(t) = exp{2e* – 2} and that of Y by My(t) = fet +. Assuming that X and Y are independent, find (a) P{X + Y = 2). (b) P{XY = 0}. (c) E(XY).
(1 point) Suppose that the moment generating function of a random variable X is My(t) = exp(4e – 4) and that of a random variable Y is My(t) = ( oer + 3)''. If X and Y are independent, find each of the following. (a) P{X + Y = 2} = (b) P{XY = 0} = (c) E[XY] = (d) E[(X+Y)?] =