Final answer:
To find the density function of the product of two independent exponential random variables, you can use the method of transformations. However, without more information about the specific distributions of the random variables, we cannot determine the exact density function of their product.
Step-by-step explanation:
The density function of the product of two independent exponential random variables can be found using the method of transformations.
Let's denote the product of the two random variables as W = T₁T₂. To find the density function of W, we need to find the cumulative distribution function (CDF) of W and then differentiate it to obtain the density function.
Using the method of transformations, you can express W in terms of T₁ and T₂ as follows: W = exp(ln(T₁T₂)). Now, to find the CDF of W, you need to find its distribution function:
F(w) = P(W ≤ w) = P(exp(ln(T₁T₂)) ≤ w) = P(ln(T₁T₂) ≤ ln(w)) = P(ln(T₁) + ln(T₂) ≤ ln(w)) = P(T₁ + T₂ ≤ ln(w)).
To find the density function, we differentiate the CDF with respect to w:
f(w) = dF(w)/dw = d/dw P(T₁ + T₂ ≤ ln(w)).
Unfortunately, we don't have enough information about the specific distributions of T₁ and T₂ to find the exact density function of their product. However, if we know the parameters λ₁ and λ₂ of the exponential distributions, we can find the joint density function of T₁ and T₂ and then use that to find the density function of their product.