Y_{n} \leq z

occurs precisely when every one of the first

n

levels is at most

z

P (Y_{n} \leq z) = F (z)^{n}, z \geq 0.

For any nonnegative random variable, the pointwise identity

Y_{n} = \int_{0}^{\infty} 1_{{Y_{n} > z}} d z

and the Tonelli theorem allow interchange of expectation and integral. Hence

E Y_{n} = \int_{0}^{\infty} P (Y_{n} > z) d z = \int_{0}^{\infty} [1 - F (z)^{n}] d z .

This identity holds as an extended nonnegative integral; finiteness of the mean is not guaranteed by continuity of

F

 Ancestors (10)

 Discussion (0) New discussion

There are no discussions about this article yet.

There are currently no matching articles.