Find the all the values of p for which both р and 1() converge. 18 1 10⁰0₁ 2n=1 n³p Op < / or p > 3 0 < p < 3 0₂ / < p < 3 O-2 < p < 2 ∞ In=
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- If n is 30 and p is 0.04. using the Poisson approximation to the Binomial, the P(X < 2) is equal to a 0.4101 b. 0 .2103 C 0.0867 d. 0 .8795Show that for the simple random walk starting at zero,P(first visit to S^(2n) occurs at t = 2k) = P(S_(2k) = 0) · P(S_(2n−2k) = 0)for 0 ≤ k ≤ n.Let X1, X2, ... , Xn be a random sample from N(μ, σ2). Find the Moment Generating Function of X̅. If n = 16 and σ = 2, compute P(-1 ≤ X̅ - μ ≤ 1).
- determine the radius and convergence of E n=1 5xn/3n2The starting time of a class is uniformly distributed between 10:00 and 10:05. If a student arrives early and has to wait t minutes for the class to start, then the student incurs a penalty of At, which accounts for the waste in the student's time. On the other hand, if the student arrives t minutes after the class has started, then the student incurs a penalty of A2t, which accounts for the information the student has missed. If the student arrives at x minutes after 10:00, what is the expected penalty incurred by the student? What value of x minimizes the expected penalty?Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?
- Everyday, James reverses his car from his driveway on to the road in such a way that there is a very small probability P that his car will be involved in a collision. ( i) Show that the probability that there will be no collision in a five-day week is (1−P)^5 and state one assumption that is made in your answer. (iii) Let P = 0.001. Check that the Poisson approximation can be used, and find the approximate probability that James will avoid a collision in 500 days.Show that the following series is divergentT2 #4 Apr 5 Of the international passengers arriving at an airport, 1.5% are selected for luggage inspection. Let X be the number of passengers from a random sample of 400 who are sent for luggage inspection. Use Poisson approximation to X to find the probability that between 6 and 12 passengers inclusive, are sent for luggage inspection.
- A time series with a periodic component can be constructed fromxt = U1 sin(2πω0t) + U2 cos(2πω0t),where U1 and U2 are independent random variables with zero means andE(U21 ) = E(U22 ) = σ2. The constant ω0 determines the period or time ittakes the process to make one complete cycle. Show that this series is weaklystationary with autocovariance functionγ(h) = σ2 cos(2πω0hConsider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θI got MLE = n/-∑logXi and UMVUE = (n-1)/∑logXi. Need help in proving consistencySuppose N = 10 and r = 3. Compute the hypergeometric probabilities for the following values of n and x. n = 4, x = 1. n = 2, x = 2 n = 2, x = 0.
