Problem 4: 10 points A random variable N has probability mass function, P[N = n] =...
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 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
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.
Suppose that Y is a random variable with the probability mass function, 2 k PſY = k] = nom 1, for k=0, 1, ..., n - 1, n (n − 1)? where n > 2. 1. Derive the expected value of Y. 2. Evaluate the second moment of Y. 3. Determine the variance of Y.
discrete random variable has probability mass function, P(X = n) = ?1?n. ? 1, forxeven Let Y = −1, for x odd Find the expected value of Y ; (E[y]). probability function mass A discrete random variable has P ( X = n) = (3) for x Y = { for Find the expected value of Y CE(y)] Let even x odd
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]
1. Suppose the random variable X has the following probability density function: Problem Set: 1. Suppose the random variable X has the following probability density function: p(x) = fcx 0sxs2 10 otherwise. ] Note this probability density function is also of the form of an unknown parameter c. (a) Determine the value of c that makes this a valid probability density function. (b) Determine the expected value of X, E[X]. (c) Determine the variance of X, V(X).
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.
1. 20 points Let X be a random variable with the following probability density function: f(x)--e+1" with ? > 0, ? > 0, constants x > ?, (a) 5 points Find the value of constant c that makes f(x) a valid probability mass function. (b) 5 points Find the cumulative distribution function (CDF) of X.
random probability course Problem 1: [60 points) A discrete random variable has the following probability mass function 2s+ 1,2,3 Find: (a) P(x <2) (b) P(l s X<3) (c) E(x) (d) Ea/x) Problem 2: [40 points] Calls to an airline reservation system have a probability of 0.7 of connecting successfully (i.e., not obtaining a busy tone). Assume that 8 independent calls are placed to the airline. (a) What is the probability that at least one call will connect successfully? (b) What...