The amount of time (in minutes) that an executive of a certain firm talks on the telephone is a random variable having the probability density
With reference to part (b) of Exercise 4.60, find the expected length of one of these telephone conversations that has lasted at least 1 minute.
Answer to relevant Questions(a) If X takes on the values 0, 1, 2, and 3 with probabilities 1/125, 12/125, 48/125 , and 64/125 , find E(X) and E(X2). (b) Use the results of part (a) to determine the value of E[(3X + 2)2]. If we let a = – µ in the first part of Theorem 4.10 on page 128, where µ is the mean of X, we get (a) Show that the rth derivative of MX – µ(t) with respect to t at t = 0 gives the rth moment about the mean of X. ...Use the moment–generating function derived in Exercise 5.20 to show that for the geometric distribution, µ = 1/θ and σ2 = 1 – θ / θ2. When calculating all the values of a Poisson distribution, the work can often be simplified by first calculating p(0;λ) and then using the recursion formula Verify this formula and use it and e–2 = 0.1353 to verify the ...Use Theorem 5.9 to find the moment– generating function of Y = X – λ, where X is a random variable having the Poisson distribution with the parameter λ, and use it to verify that σ2Y = λ.
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