How do I simplify the sum of the image?

Question

How do I simplify the sum of the image? I am studying the probability function of order statics X(i). It differentiates the distribution function, but I don't know how to simplify the sum to obtain the result of the image.

Answer 1

The final equation is suggesting that the first term of the first sum is the only man standing. That's indeed the case. For the first sum do the following: \begin{align*} \sum_{j=i}^n\dbinom{n}{j}jf(x)(F(x))^{j-1}(1-F(x))^{n-j} &= \dbinom{n}{i}if(x)(F(x))^{i-1}(1-F(x))^{n-i}\\ &\quad + \sum_{j=i}^n\dbinom{n}{j+1}(j+1)f(x)(F(x))^{j}(1-F(x))^{n-j-1}. \end{align*} (Note the subtle change of index in the sum). Now you just need to prove that $$ \dbinom{n}{j+1}(j+1)=\dbinom{n}{j}(n-j). $$