Question

(1) Let X have the Laplace density λe−λ|x|/2, for x∈ℜ and λ>0. Provide the justifications to the following steps that show that its characteristic function is λ2/(λ2+t2).

Step 1. Why is the following true?

.∫0∞cos⁡(ty)e−λydy=λλ2+t2.

Step 2. Why is the following true?

.∫−∞∞sin⁡(ty)e−λ|y|dy=0.

Step 3. Why are the above two results are enough to complete the solution?

The following ``answers'' have been proposed. Please read carefully and choose the most complete and accurate choice.

(a) Step 1 follows by converting to polar coordinates and one integration by parts. Step 2 follows by Taylor expansion of the integrand and Step 3 follows by Fubini's theorem.
(b) Steps 1 and 2 both follow by Riemann-Lebesgue lemma. Step 3 follows by Fubini's theorem.
(c) Step 1 is by two integrations by parts. Step 2 follows by Riemann-Lebesgue lemma. Step 3 follows by Fubini's theorem.
(d) Step 1 follows by two integrations by parts. Step 2 is the consequence of odd integrable function. Step 3 follows by the linearity of the Lebesgue integral.
(e) None of the above.

The correct answer is

Answer 1