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- If X1, X2, ... , Xk have the multinomial distribution of Definition 8, show that the mean of the marginal distribu-tion of Xi is nθi for i = 1, 2, ... , k.A harried passenger will be several minutes late for a scheduled 10 A.M. flight to NYC. Nevertheless, he might still make the flight, since boarding is always allowed until 10:10 A.M., and boarding is sometimes permitted up to 10:30 AM. Assuming the end time of the boarding interval is uniformly distributed over the above limits, find the probability that the passenger will make his flight, assuming he arrives at the boarding gate at 10:25.X is uniformly distributed on the interval (3, 15). What is the probability that X is at least 5?
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- Let X represent a continuous random variable with a uniform distribution over the interval from 0 to 4. What is P(X < 0.9)?Can you help please: Suppose that X ∼ Geom(p). In chapter 2, we computed the CDF of X as FX(x) = 1 − (1 − p)x,for x = 1, 2, · · · . If p = λ/n, show that Y = X/n converges to an exponential distribution withparameter λ as n → ∞.Assume that X = (X1,..., Xn) is a vector of independent observations of the bernoulli distribution X ∼ Be (θ) where θ ∈ [0, 1] is unknown. It is assumed that θ has the a priori distribution Beta (α, β), with α = 4 and β = 2. Determine the a posteriori distribution of θ given that you have n = 10 and the observation vector x = (1, 0, 1, 0, 1, 0, 0, 1, 0, 0).