Question

Escalate - Show every mynute steps in DETAILED! Explain & SHOW how the LIMITS of integration & DOMAIN are found.

a) Let X be a random variable with uniform distribution on the interval a,b). Find its moment generating function. 2n b) The random variable X has the following moments: E(X") = n = 1, 2, .... n+1 Find the moment generating function of X and identify the distribution of X. Hint: Use the Taylor expansion of etx to find the moment generating function, and use problem a) to identify the distribution.

Answer 1

Part ca) x ~ uniform (a,b). The probability density function of f(x) = I asx sb. f is: ba generating function Lit Mx ( t) be the moment (MGF) of X. By definition, Mx (t) = E[ etx] - tx acanda f (x) dx tx boa on S et de btlo ) toep t(bra)

PUz Mx Ct) is Note mell that defined the for expression for any t to. 0.X For t=0 Mx (t) = Mx(0) = Elex) = E(1) a 1, So, MGF of x is on tb ta - et to t to Mx (t) = (6-a) 1 t=0 Part (6) & Suppose X be a RV haning moments E(X") = 2 , n=1,2,3,.... nal we The need to find the map of x. Taylor series expansion of eth is given by: & t tR,

A e tx is a expectation on convergent power forts, taking both sides, we get Mx 623 Ele) - ELE ] L using the expression E(X) = 2 forra' Ico + S (2 pla t. 2 to -- PERVO

me see that the form of this Mar is exactly the same obtained in part (a) with a co &b=2 Thus, by uniqueness Property of Mat, x ~ Uniform (0,2).
In part (a), the expression for moment generating function of Uniform(a,b) distribution is derived.

In part (b), we derive the MGF of the Random Variable X using the expression for raw moments and taylor series expansion of exp(tX) . Hence, we find that the expression matches with that found in part (a). Using Uniqueness Property, we can conclude that X follows Uniform(0,2)