Problem 1: 10 points Suppose thatY is a random variable with the probability mass function, PY...
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.
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.
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.
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).
Problem1 Random variable Y has a probability mass function (pmf) as py(y) = a) Find the value of the constant c ,y=1,2,3 , y =-1,-2,-3 0 otherwise b) Now that the constant c is determined, find (G) Probability of Y 1 (ii) Probability of Y<1
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...
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity λ-1. Given L-u, a random variable Y has the Poisson distribution with parameter - u. 1. Derive the marginal distribution of Y and evaluate probabilities, PY=n] , for n = 0,1,2, 2. Find the expectation of Y, that is E Y 3. Find the variance of Y, that is Var Y
4.8. Let Z be a random variable with the geometric probability mass function where 0 < π < 1. (a) Show that Z has a constant failure rate in the sense that PriZ kZk1 T for k 0, 1,.... (b) Suppose Z' is a discrete random variable whose possible values are 0, 1, and for which Pr(Z'=KZ2k} = 1-π for k 0,1,.... Show that the probability mass function for Z' is p(k).
3. (10 points) Let X be continuous random variable with probability density function: fx(x) = 7x2 for 1<<2 Compute the expectation and variance of X 4. (10 points) Let X be a discrete random variable uniformly distributed on the integers 1.... , n and Y on the integers 1,...,m. Where 0 < n S m are integers. Assume X and Y are independent. Compute the probability X-Y. Compute E[x-Y.
Problem 2 [17 points]. Transformations! a) (5 points) Suppose the time, W, it takes to complete a technical task at a workshop has probability density function -w/2 f(w)y 0, 0, otherwise Using the appropriate transformation methods, find the density function for the a time it takes two workers to complete this technical task: S Wi + Ws b) (5 points) Derive the moment generating function of a standard normal randon variable. Use point form to explain each step in your...