Let X ∼ Unif[0, 1]. Find a linear function Y = g(X) such that the interval [0, 1] is transformed into [–5, 5]. Use the relationship for linear functions M_{aX + b}(t) = e^btM_X(at) to obtain the mgf of Y from the mgf of X. Compare your answer with the result of Exercise 32, and use this to obtain the pdf of Y.

Data From Exercise 32 

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- Let X have a uniform distribution on the interval [A, B], so its pdf is f(x) = 1/(B − A), A ≤ x ≤ B, f(x) = 0 otherwise. Show that the moment generating function of X is Mx(t) = eBt - eAt (B - A)t t#0