A question regarding exponential distribution Charlie and Bella and their friends Mark and Leonard each have a toy. Each toy breaks at a time that is exponentially distributed with expectation $24$ hours. Assume the toys are independent of each other.

What is the cumulative distribution function of the time until Charlie's toy fails?

Let $X$ be the time until Charlie's toy fails. $E[X]=\frac{1}{\lambda}=24$. Then $\lambda=\frac{1}{24}$. And so the cumulative distribution function is
$$F(x) = \begin{cases}
1-e^{-x/24}& \text{ if } x \ge 0, \\ 
0 & \text{ if } x \lt 0.
\end{cases}$$

Let $T$ be the time until all the toys have failed. Compute the cumulative distribution function of $T$.

The toys are independent of each other, so $E[T]=96$. Then $\lambda=\frac{1}{96}$. Hence the cumulative distribution function of $T$ is 
$$F(t) = \begin{cases}
1-e^{-x/96}& \text{ if } t \ge 0, \\ 
0 & \text{ if } t \lt 0.
\end{cases}$$

Let $S$ be the time until the first toy has failed. Compute the cumulative distribution function of $S$.

Are the attempts above correct? How do I do the last one? Thank you.
 A: The first part is solved correctly.
Let $T$ be the time until all the toys have failed. The probability that $T\le t$ is the probability that all the toys have failed by time $t$. By independence this is the fourth power of the cdf you calculated in the first part. In symbols,
$$F_T(t)=\left(1-e^{-t/24}\right)^4.$$
The time $T$ is usually called $\max(X_1, X_2,X_3,X_4)$, where the $X_i$ are the lifetimes of the toys of the various people.
For the last, let $S$ be the time until the first failure. The probability this is $\gt s$ is the probability all the toys are alive at time $s$. This is $(e^{-s/24})^4$. So
$$F_S(s)=1-\left(e^{-s/24}\right)^4.$$
The time $s$ is usually called $\min(X_1, X_2,X_3,X_4)$.
Remark: In your solution of the second problem, you asserted that the mean of $T$ is the sum of the means of the $X_i$. There is no reason to think that. The mean of the sum of the $X_i$ is the sum of the means, but $T$ is not the sum of the $X_i$. You also assumed that $T$ has exponential distribution. It doesn't. However, $S$ does.
