6. In the text we noted that

$$E[\sum_{i=1}^{n} X_i] = \sum_{i=1}^{n} E[X_i]$$

when the X, are all nonnegative random variables. Since an integral is a limit of sums, one might expect that

$$E[\int_{0}^{∞} X(t) dt] = \int_{0}^{∞} E[X(t)] dt$$

whenever X(1), 0, are all nonnegative random variables; and this result is indeed true. Use it to give another proof of the result that, for a nonnegative random variable X,

$$E[X) = \int_{0}^{∞} P(X>1) dt$$

HINT: Define, for each nonnegative t, the random variable X(1) by

$$X(t) = \begin{cases}

1 &\text{if } t < X\\

0 &\text{if } t \ge X

\end{cases}$$

Now relate X(1) dt to X.