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# Q5 - functions ﬁnd the m.g.f of x 1 ∼ e − x 1 and x 2...

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EXERCISES IN STATISTICS Series A, No. 5 1. Find the moment generating function of x f ( x ) = 1, where 0 < x < 1, and thereby confirm that E ( x ) = 1 2 and V ( x ) = 1 12 . 2. Find the moment generating function of x f ( x ) = ae ax ; x 0. 3. Prove that x f ( x ) = xe x ; x 0 has a moment generating function of 1 / (1 t ) 2 . Hint: Use the change of variable technique to integrate with respect to w = x (1 t ) instead of x . 4. Using the theorem that the moment generating function of a sum of in- dependent variables is the product of their individual moment generating
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Unformatted text preview: functions, ﬁnd the m.g.f. of x 1 ∼ e − x 1 and x 2 ∼ e − x 2 when x 1 , x 2 ≥ 0 are independent. Can you identify the p.d.f. of f ( x 1 + x 2 ) from this m.g.f.? 5. Find the moment generating function of the point binomial f ( x ) = p x (1 − p ) 1 − x where x = 0 , 1. What is the relationship between this and the m.g.f. of the binomial distribution ?...
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