Mary bought 10 bulbs, each with an exponential lifetime of 500 hours. Let $ T $ be the time at which they all burn out. What's the cdf and pdf of $ T $?
Here is what I started with:
pdf of lifetime of each bulb: $ f(t) = \frac{1}{500} e^{-t /500} $
joint pdf of the lifetime of all 10 bulbs:
$$
f(t_1, t_2, …, t_{10}) =f(t_1) f(t_2)…f(t_{10}) =\frac{1}{500^{10}} e^{-(t_1 + t_2 +… +t_{10}) /500}
$$
How do I get to the pdf and cdf of $ T $ (the time at which they all burn out) based on the above?
Best Answer
Let $X_1, X_2, \dotsc, X_{10}$ be the lifetimes of the 10 bulbs. The time $T$ at which they all have burned out is the maximum of these lifetimes i.e. $T=\max(X_1,\dotsc, X_{10})$. To compute the cdf of $T$ observe that (for $t\geq 0$) $$ T\leq t\iff X_1\leq t,\dotsc, X_{10}\leq t $$ so $$ P(T\leq t)=P(X_1\leq t,\dotsc, X_{10}\leq t)\stackrel{(\star)}{=}\prod_{i=1}^{10}P(X_1\leq t)=(1-e^{-t/500})^{10} $$ where in $(\star)$ we used independence and identically distributed assumption. Differentiate to find the density.