Hey I am getting ready for my final exam and I'm having trouble figuring out this practice question:
Let X be a random variable that takes values in {0,1,2,3,...}. It is known that:
E(X) = $\displaystyle\sum_{k=1}^{\infty} k * Pr(X = k)$
Define an infinite matrix and use it to prove that:
E(X) = $\displaystyle\sum_{k=1}^{\infty}Pr(X \ge k)$
I am thinking that we would need to add up all sums of columns but I'm not quite sure where to go after that.
I'm assuming you start with the corrected version
$E(X) = \displaystyle\sum_{k=1}^{\infty}P(X \ge k)$
You can write $Pr(X \ge k)$ as $\sum_{j=k}^\infty P(X =j)$. Now we have
$E(X) = \displaystyle\sum_{k=1}^{\infty} \displaystyle\sum_{j=k}^{\infty} P(X =j)$.
We're free to choose the order of in the summation (think of instead of summing column-wise than row-wise, do the other way around). So we can write
$E(X) = \displaystyle\sum_{j=1}^{\infty} \displaystyle\sum_{k=1}^{j} P(X =j)$. Since the summand is constant wrt $k$ this can be rewritten simply
$E(X) = \displaystyle\sum_{j=1}^\infty jP(X = j)$