Problem 8: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the...
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. 1. Determine conditional density of the residual lifetime, T-u, given that T>u 2. Find conditional expectation, EITIT>] Solution
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. I. Determine conditional density of the residual lifetime, T-u, given that T 〉 u. 2. Find conditional expectation, E TT>u
Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. 1. Determine conditional density of the residual lifetime, T - u, given that T u 2. Find conditional expectation, E [T|T > u]
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
5. A light bulb has a lifetime that is exponentially distributed with rate parameter λ-5. Let L be a random variable denoting the sum of the lifetimes of 50 such bulbs. Assume that the bulbs are independent. (a) Compute E[L] and Var(L). b) Use the Central Limit Theorem to approximate P(8 < L < 12 ( ). (c) Use the Central Limit Theorem to find an interval (a,b), centered at ELLI, such that Pa KL b) 0.95. That is, your...
Assumptions from problem #2 Problem 3: 10 points Continue with the same assumptions as in Problem 2. Recall that a random variable, Z, has the Gamma distribution with the density: fz (z) = λ2 z exp[-λ z] for z > 0, and fz(z) = 0, elsewhere. Conditionally given Z = z, a random variable, U, is uniformly distributed over the interval, (0, z) 1. Find conditional expectation. EZIU = ul. 2. Find conditional variance, VARZİU-ul 3. Find conditional expectation, E...
Problem 10: 10 points Assume that the distribution of the lifetime of a certain appliance is given. Knowing that the device has served five years, find the chance that it will serve 5 more years 1. Assume that the lifetime, T, is exponentially distributed, with the mean equal to 25 years. 2. How will your answer change if the lifetime (T) is uniformly distributed over the interval, (0,25)?
Problem 5: 10 points Assume that a discrete random variable, N, is Poisson distributed with the rate, λ = 3. Given N = n, the random variable, X, conditionally has the binomial distribution, Bin [N +1, 0.4] 1. Evaluate the marginal expectation of X. 2. Evaluate the marginal variance of X
Problem 1 [20 points X is an exponentially distributed random variable with parameter A. a, b with b >a 0 are real numbers. Find PLX > E [a, b))