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- The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.Let m(t) be the moment generating function of a random variable X. Show that the random variable W = 10X is m(10t). What is the moment generating function of Z = X-5 in terms of m(t)?Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50
- Use the moment generating function to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n.Find the distribution of Y = X1 + · · · + Xn.If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?Use the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.
- Suppose that the random variable X is continuous and takes its values uniformly over the interval from 0 to 2. What is P{X = 1.5 or X = 0.4}?Suppose that X has a Poisson distribution with h = 80. Find theprobability P(x is less than or equal to 80)Let Mx, y be the moment generating function of random variables that are not independent of X and Y. Which of the following / which are not the properties of the function Mx, y?
- 2.) Use the moment generating function you just proved to verify that for a Poisson distribution, E(X) = λ and Var(X) = λ.T2 #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.Let X be a random variable with pdf given by fX(x) = 1/[π(1 + x2)] for all real number x. Prove that X and 1/X are identically distributed by showing that they have the same probability distribution.
