Problem 1 [20 points X is an exponentially distributed random variable with parameter A. a, b...
Let X be an exponentially distributed random variable with parameter λ. Prove that P(X > s + tK > t) P(X > s) for any S,12 0
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.
2. Let X be an exponentially distributed random variable with parameter 1 = 2. Determine P(X > 4). 3. Let X be a continuous random variable that only takes on values in the interval [0, 1]. The cumulative distribution function of X is given by: F(x) = 2x² – x4 for 0 sxsl. (1) (a) How do we know F(x) is a valid cumulative distribution function? (b) Use F(x) to compute P(i sX så)? (c) What is the probability density...
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
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 ]
I. Let Y be an exponentially distributed random variable with parameter λ Compute the cdf and the pdf for the random variable X-e
Let X be an exponentially distributed random variable with parameter value 2. a) Use the markov inequality to estimate P(X >= 3). b) Use the chebyshev inequaity to estimate P(X >= 3). c) Calculate P(X >= 3) exactly.
a) Let T be an exponentially distributed random variable with parameter l= 1. Let U be a uniformly distributed random variable. Use inversion to show how to calculate samples {tı, t2, ..} from samples {U1, U2, ..} of U. b) Use any software of your choice to estimate by Monte Carlo simulation: E[sin(tanh, LT))
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~ Exp(A), so that its probability density function (pdf) of X is SO e /A fx(x) | 0, (2) (a) Let Y log(X. When A = 1, (i) Show that the pdf of Y is fr(y) = e (u+e-") (ii) Derive the moment generating function of Y, My(t), and give the values of t such that My(t) is well defined. (b) Suppose that Xi, i...