Problem 7: 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. 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 8: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ 〉 0. 1. Find conditional variance, Var TIT〉 u] . 2. Find conditional second moment, E T IT ]
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 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))
Problem 7: 10 points Assume that the inter-arrival times, S the renewal process, j21, are independent and exponentially distributed. Consider N = {N(t): t 0), defined as before: 1. Derive the conditional density of W2, given Ws<t< Wo 2. Derive the conditional expectation of (Ws - W2), given Ws<t< Wo 3. Derive the marginal expectation of (W1-W2), assuming that the rate is
3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and further suppose that U is a uniformly distributed random variable between 0 and 1 that is independent from X and Y. Calculate Pr(X<U< Y) and estimate numerically (based on a visual plot, for example) the value of λ that maximizes this probability.
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
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