10) (11) Let X and Y be 2 independent random variables. Suppose X ~ Gamma(0, 38)...
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 .
Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint: Determine first the probability distribution of T -X +Y + Z using the moment generating function method. Moment generating function for Poisson random variable is given in earlier lecture notes) Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint:...
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...
3Y 2 1. (20 points) Suppose that X and Y independent random variables. Let W 2x (a) Consider the following probability distribution of a discrete random variable X: 12 P(X) 00.7 0.3 X Compute the mean and variance of X (b) Use your answers in part (a). If E(Y)=-3 and V(Y)= 1, what are E(W) and V (W)?
Let Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter p. Suppose that Y, X1 and X2 are independent. Proof using the de finition of distribution function that the the distribution function of Z =Y Xit(1-Y)X2 is F = pF14(1-p)F2 Don't use generatinq moment functions, characteristic functions) Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter...
Suppose X, Y and Z are three different random variables. Let X obey Bernoulli Distribution. The probability distribution function is p(x) = Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y ∼ N(0, 1). X and Y are independent. Meanwhile, let Z = XY . (a) What is the Expectation (mean value) of X? (b) Are Y and Z independent? (Just clarify, do not need to prove) (c) Show that Z is also a standard...
Let X and Y be independent random variables with pdf 2-y , 0sys2 2 f(x) 0, otherwise 0, otherwise ) Find E(XY) b) Find Var (2X+3Y)
Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y follows a geometric distribution with parameter p. Assume that X takes only nonnegative integer values, and let Gx(z) be the probability generating function of X. (We make no additional assumptions about the distribution of X.) Show that P(X<Y) = Gx(1- p). Clearly indicate the step(s) in your argument that use the assumption that X and Y are independent.
Let X, Y be independent random variables with E[X] = E[Y] = 0 and ox = Oy = 5. Then Var(2x+3Y) = 1. True False
Let X, Y be independent random variables with E[X] = E[Y] = 0 and ox = oy = 5. Then Var(2x +3Y) = 1. True False