You have a bag full of 100 LED and 100 CFL bulbs. The lifetime of a LED light bulb in hours is distributed as Exp(50000) whereas that of a CFL bulb is distributed as Exp(10000). Let X denote the lifetime of a bulb randomly chosen from the bag. Calculate the following:
(a) var(X)
I'm just not sure how to start off here. I know covariance will come into play at some point, but I just don't know how to begin this problem. Thank you!
Let $U$ be the lifetime of an LED, and $V$ the life of a CFL. Then $$E(X)=\frac{1}{2}E(U)+\frac{1}{2}E(V).$$ Now we know $E(X)$.
Next we find $E(X^2)$. This is $\frac{1}{2}E(U^2)+\frac{1}{2}E(V^2)$.
Note that by properties of the exponential we know that $E(U^2)=2(50000)^2$ and $E(V^2)=2(10000)^2$.
Now we know $E(X^2)$. Finally, $\text{Var}(X)=E(X^2)-(E(X))^2$.