Define the moment generating function of a random variable and state its pronerties.
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- 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?Compute the probability that a Random Walk ever returns to the origin and the expected time to the first return.If X, Y are normally distributed independent random variables, then show that W = 2X - Y is normally distributed. Do not use moment generating function, only use convolution formula.
- 2.) Use the moment generating function you just proved to verify that for a Poisson distribution, E(X) = λ and Var(X) = λ.Decide whether the random variable X is discrete or continuous. X represents the tension at which a randomly selected guitars strings have been strong.Derive the moment generating function (MGF) of a random variable ?.
- Show that the t-distribution with r = 1 degree of freedom and the Cauchydistribution are the same.Let X1, X2, ... Xn random variables be independent random variables with a Poisson distribution whose parameters are l1, l2, ... ln, respectively. Which of the following is the moment generating function of the random variable Z defined as (the little image)?If X has an F distribution with ν1 and ν2 degrees offreedom, show that Y = 1Xhas an F distribution with ν2 and ν1 degrees of freedom.
- Let X be a continuous random variable with pdfShow that if the limiting conditions n→q, θ →0, while nθ remains constant, are applied to the moment-generating function of the binomial distribution, we get the moment-generating function of the Poissondistribution.The number of passengers willing to travel with a certain train is as- sumed to be a Poisson variabel with parameter µ = 300. How many seats should the train have if the probability for an over-booked train should be at most 1%?