Question

let s1 and s2 be exponential i.i.d. random variables with rate 2. find the probability density function of y "" s1 ` s2.

Answer 1

To find the pdf of the variable Y = S1 * S2, where S1 and S2 are exponential i.i.d. random variables with rate 2, we need to find the CDF of Y and differentiate it to obtain the pdf of Y. The CDF of Y is given by P(Y <= y) = P(S1 * S2 <= y) = P(S1 <= y/S2). Since S1 and S2 are independent, the joint distribution of S1 and S2 is the product of their individual distributions.

Step-by-step explanation:

To find the probability density function (pdf) of the variable Y = S1 * S2, where S1 and S2 are exponential i.i.d. random variables with rate 2, we first need to find the cumulative distribution function (CDF) of Y and then differentiate it to obtain the pdf of Y.

The CDF of Y is given by:

FY(y) = P(Y <= y) = P(S1 * S2 <= y) = P(S1 <= y/S2)

Since S1 and S2 are independent, the joint distribution of S1 and S2 is the product of their individual distributions:

P(S1 <= y/S2) = ∫ P(S1 <= y/x) * fS2(x) dx, where fS2(x) is the pdf of S2.

Since S1 and S2 are exponential random variables with rate 2, their pdfs are given by:

fS1(x) = 2 * e^(-2x)

fS2(x) = 2 * e^(-2x)

Substituting these values into the integral, we get: