Question

The Laplace distribution (also known as the double-exponential distribution) is a continuous distribution with location parameter m ER and density given by fm (x) = fe e-ml. Let X denote a Laplace random variable with location parameter set to be m = 0. What is E[X]? Does the variance o2 = E[(x – E[X])21 exist? Yes No Which of the following are true about X? (Choose all that apply.) Hint: The function chez is integrable, i.e. L ke-12 dc is finite for all k. The distribution of X is symmetric in the sense that X and - X have the same distribution. The function In fm (2) has a continuous first derivative. For any integer k > 0, the k-th moment E[Xk] exists.

Answer 1

1.For a double exponential distribution, expectation is the location parameter.

Here location parameter is m=0

Hence E(x) = m= 0.

2. For double exponential distribution, we know that the variance exists and is used in the scale parameter. Hence yes, the variance exists.

Hence correct answer is Yes.

3. Clearly the PDF of X for m=0 , is symmtric around 0 as for +x and -x , the PDF remains the same. Hence 1st option is correct.

2nd option is incorrect as logarith of pdf is ln(1/2)-|x| and this does not has continuous first Oder derivative.

We know that MGF of double exponential distribution exists and hence raw moments of all order exist. Hence 3rd option is correct.

Hence 1st and 3rd options are correct.

Hope the solution helps. Thank you.

(Please comment if you need further help. You may consult any book or internet for details regarding this distribution)