Consider the random variable X whose density is given by f(x) = (x - 3)^2/5 x =

Question

Consider the random variable X whose density is given by f(x) = (x - 3)^2/5 x = 3, 4, 5 (a) Verify that this function is a density for a discrete random variable. b) Find E[X] directly. That is, evaluate Sigma_all x xf(x). (c) Find the moment generating function for X. (d) Use the moment generating function to find E[X], thus verifying your answer to part (b) of this exercise. (e) Find E[X^2] directly. That is, evaluate Sigma_all x x^2f(x). (f) Use the moment generating function to find E[X^2], thus verifying your answer to part (e) of this exercise. (g) Find sigma^2 and sigma. A discrete random variable has moment generating function m_x(t) = e^2(e^- 1) (a) Find E[X]. (b) Find E[X^2]. (c) Find Sigma^2 and sigma.

Explanation / Answer

m(t) = e^(2*(e^t -1))

m’(t) = 2 e^(t+ 2e^t -2)

m’’(t) = (4 e^t + 2) e^(t+2e^t -2)

a)

E[X] = m’(0)

= 2 * e^(0 + 2 -2)

= 2

b)

E[X^2] = m’’(0)

= (4 +2) e^0

= 6

c)

Var(x) = E(x^2) - E(x)^2

= 6 - 2^2

= 2

standard deviation = sqrt(2) = 1.414

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