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- 5. Suppose that a stick of length 1 is broked at a point U uniformly distributed on (0,1). Let L be the length of the longer of the two resulting pieces. 0 (a) Find P (L>). (b) 1 Find the cumulative distribution function of L.Let X; € {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid fx\p(x |p) = (1 – p)*-'p x-1 X1, X2, for 0A.2) The time between successive customers coming to the market is assumed to have Exponential distribution with parameter lambda. a) If X1, X2. ..., Xp are the times, in minutes, between Successive customers selected randomly, estimate the parameter of the distribution. b) The randomly selected 15 times between successive customers are found as 1.8, 1.2, 0.8, 1.4, 1.2, 0.9, 0.6, 1.2, 1.2, 0.8, 1.5, 1.8, 0.9, 1.5 and 0.6 mins. Estimate the mean time between successive customers, and write down the distribution function. c) In order to estimate the distribution parameter with 0.4 error and 4% risk, find the minimum sample size.Could you solve (g),(h)? Thank you.Assume that X is a random variable whose conditional distribution given the variable Y is poisson P (X | Y) = Po (Y). Suppose further that Y has a gamma distribution Y ∼ Gamma (1, 1). (a) Determine the value E (XY). (b) Determine the conditional distribution P (Y | X).Let X1, X2, .., Xn be a sample from a Continuous Unif(u - Ō, µ + d) distribution. Please find the moment estimators (MoM) for µ and ō. [Note] ð > 0. .... (b-a)? [Hint]: If X - Unif (a, b), then E(X)="", V(X)="1" a+l(e) If X and Y are iid with X - Poisson (1), then 2X +Y ~ Poisson (3). true_ L; false (f) If X and Y are iid with a common discrete distribution, then P(X < Y) < 1/2. true false (g) If X and Y are iid with X ~ Poisson (1), then Var(2X – Y) = 5. L; false trueSuppose that a random sample of sizen is taken from a Poisson distribution for which the value of the mean e is unknown, and the prior distribution of e is a gamma distribution for which the mean is Po. Show that the mean of the posterior distribution of e will be a weighted average having the form Y,X, + (1– Yn)Ho, and show that yn →1 as n- *.Let X; € {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2,..., Xn fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1,2, 3, ...} and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fp|X, (p | Xn).The random variable X has expected value E(X)=4 and Var(X)=4. Here, we want to approximate the probability that X lies in the interval [3.4,5.4] using a transformation to a standard normal distribution Z. Then we will need to calculate the probability that Z lies in the interval [z1,z2] where z1= z2= Suggest z1 and z2 Do NOT attempt to make any sort of continuity correction.Let Y1, Y2,., Ya be a collection of independent random variables with distribution function y 8 Show that Y converges in probability to a constant, and provide that constant. 1SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON