I want to find PDF by differentiating CDF and then from PDF, expected values of the following problem. .

Question

Three light bulbs have independent exponentially distributed lifetimes with a common parameter $\lambda$. What is the probability distributed function and expected value of the time until the last bulb burns out?

Answer 1

Answer is as follows:

$F_{Y}(y) = P(\max(X_1,X_2,X_3)<y) = P(X_1<y)P(X_3<y)P(X_3<y) = (1-e^{-\lambda y})^3$

Now for PDF ($f_{Y}(y)$), we have to diffrentiated CDF ($F_{Y}(y)$):

$\frac{d F_{Y}(y)}{dy} = f_{Y}(y) = \int_{0}^{\infty}3\lambda e^{-\lambda y}(1-e^{-\lambda y})^2 dy$

For expected value E[Y], we have the following formula:

$E[Y] = \int_{0}^{\infty} y 3\lambda e^{-\lambda y}(1-e^{-\lambda y})^2 dy = \frac{11}{6 \lambda}$