Suppose that for a random variable X, X, E(X") 2", n generating function and the probability...
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...
. If X is a random variable with probability generating function Px(2)ze-1-*), then , then (a) Calculate the mean and variance of X. (b) What is the distribution of X? Hence, give the mass function of X. (Hint: Think about your answer to 2(d).)
Q 2. The probability density function of the continuous random variable X is given by Shell, -<< 0. elsewhere. f(x) = {&e*, -40<3<20 (a) Derive the moment generating function of the continuous random variable X. (b) Use the moment generating function in (a) to find the mean and variance of X.
6. Suppose the moment generating function of a random variable X is My(t) = (1 – 2+)-3, fort € (-1/2,1/2) Use this to determine the mean and variance of X.
Problem 2: 10 points Suppose that X is a random variable with the probability mass function, 10-k), n (n -1) for k=1, 2, ,n, where n 22. 1. Derive the expected value of X. 2. Determine the variance of X. 3. Derive the second moment of X.
(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)?] =
A5. The probability generating function of a random variable X takes the form 67,) 0.3 2 G(s) 1-07s Use this probability generating function to compute: (a) The mean and variance of X (b) The probability that X takes the value 2.
3. A random variable X has the probability mass function P(x = k) = (a > 0, k =0,1,2...). (1 + a)! Find E[X], Var(X), and the Moment generating function My(t) = E[ex]
The moment generating function of a random variable X is as follows: 1-Xt Find the probability that X is within 0.5 standard deviation from its mean.
The probability mass function of a random variable X is given by PX(n) = (c)(λ^n) / n! , n = 0, 1, 2, . . . (a) Find c (Hint: use the relationship that (SUM) n=0->∞ ( x^n / n! = e^x ) (b) Now assume λ = 2, find P(X = 0) (c) Find P(X > 3)