Suppose that Y is a random variable with the probability mass function, 2 k PſY =...
Problem 1: 10 points Suppose thatY is a random variable with the probability mass function, PY n-1) 2k_-, for k=0,1, ,n-1, where n 2. 2 1. Derive the expected value of Y. 2. Derive the second moment of Y 3. Determine the variance of Y
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.
Problem 4: 10 points A random variable N has probability mass function, P[N = n] = p(n) that satisfies the equation n+1 p(n) p(n+1) = valid for all n = 0, 1, ..... 1. Evaluate p0) using the fact that Îr(n) = 1. 2. Find expected value of N. 3. Determine the second moment of N.
Let X be a discrete random variable with probability mass function p(k) = 1/5, k = 1, 2, . . . , 5, zero elsewhere. (a) Find the moment generating function of X. (b) Use the moment generating function in (a) to determine the convolution of two identical probability mass functions given above. This is identical to asking the probability mass function of X + Y and where X and Y are independent and each has probability mass function given...
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n =...
Question 4. [5 marksi Let Xbe a random variable with probability mass function (pmf) A-p for -1, 2,... and zero elsewhere (whereq-1-p, 0 <p< (a) Find the moment generating function (mg ofX. C11 (b) Using the result in (a) or otherwise find the expected value and variance of X. C23 (c) Let X, X,., X, be independent random variables all with the pmf fix) above, and let Find the mgf and the cumulant generating function of Y.
A continuous random variable Y has density function f(y) = f'(y) = 2 · exp[-4. [y] defined for -00 < y < 0. Evaluate the cumulative distribution function for Y Consider W = |Y| and find its C.D.F. and density Determine expected value, E [Y] Derive variance, Var [Y]
Suppose X is a random variable with probability mass function f(x) = k if x = 0 k if x = 1 3k if x = 2 14/25 if x = 3 Find k. A. 11/500 B. 11/250 C. 11/125 D. 11/120 E. none of the preceding
Suppose Y is a discrete random variable with probability mass function p(y) - P(Y -y) - fory - 1,2, ..., n. Show that p(y) satisfies the conditions of a pmf.
Suppose that for a random variable X, X, E(X") 2", n generating function and the probability mass function of X Hint: Use (11.2). 1, 2, 3, . . . the moment