Problem 2. (3 marks; 3, 2) Let Yi and Y, be two independeet discrete Let...
Problem 2. (5 marks. 3, 2) Let Yi and Y2 be two independent discrete random variables such that: pi (yi) = ,--2-1, 0 and P2(U2) = 2 = 1.6 Let K = Yi + Y2. a) Find the moment generating function of Y1,Y2 and K. b) Using part a), find the probability mass function of K
Let Y1 and Y2 be two independent discrete random variables such that: p1(y1) = 1/3; y1 = -2 ,- 1, 0 p2(y2) = 1/2; y2 = 1, 6 Let K = Y1 + Y2 a) FInd the moment Generating function of Y1, Y2, and K b) find the probability mass function of K
3. Let Yİ ~Gamma ( -3,ß-3), Y ~Gamma( -5, ß-1), and W-2% + 6K. a) (9 pts) Find the moment generating function of W. Justify all steps b) (3 pts) Based on your result in part (a), what is the distribution of W(name and parameters)?
Let the random variable Y have the following probability distribution y 2 4 6 P(Y=y) 4/k 1/k 5/k find the value of k. find the moment-generating function of Y find Var(Y) using the moment generating function let W= 2Y-Y^2 +e^2*Y+7. find E(W)
7. Let Xbe a discrete random variable with probability mass function: f(x) c , x 0,1,2,3, 4 (a) Find the constant c. (b) Find the moment generating function of X. (c) Find EX based on the result of part (b)
PLEASE MAKE YOUR HAND WRITING CLEAR AND READABLE . THANK YOU! O Let X and Y be independent random variables with a discrete uniform distribution, i.e., with probability mass functions for k = 1, px(k) = py (k) =-, N. Use the addition rule for discrete random variables on page 152 to determine the probability mass function of Z -X+Y for the following two cases. a. Suppose N = 6, so that X and Y represent two throws with a...
Thus X has the probability mass function (for x1, x2, 3 E No with z +22 +3n) and moment generating function You do not need to show this.) i. Show that the moment-generating function of Y = (YĪ, defined as s given by 2 MARKS ii. Show that Yİ ~ Bi(n, 1 + 2) Hint: The probability mass function and moment-generating function of the binomial distribution can be found on the formula sheet.[3 MARKS] iii. One can also show that...
Problem 3 Let X be Uniform(0,1) and Y be Exponential (1). Assume that X and Y are independent. i. Find the PDF of Z- X +Y using convolution. ii. Find the moment generating function, øz(s), of Z. Assume that s< 0. iii. Check that the moment generating function of Z is the product of the moment gen erating functions of X and Y Problem 3 Let X be Uniform(0,1) and Y be Exponential (1). Assume that X and Y are...
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...
1. Let X and Y be two discrete random variables each with the same the possible outcomes {1,2,3} a) Construct a bivariate probability mass function Px.y : {1,2,3} x {1,2,3} + R that satisfies the following properties propeties: (i) The expectation of X is E[X] = 2.1, (ii) The conditional expectation of Y given 2 = 3 is EY 2 = 3] = 1, (iii) The correlation between X and Y is slightly positive so that 0 < corr(X,Y) <...