Thread: sum of Bernoulli trials

Hoi, i dont see why the following is true

Let ( i.i.d. Bernoulli trials)

and consider ( Then clearly )

Apparantly the distribution function F of Y is constant on the interval and satisfies
and

for it satisfies

I'm a bit bewildered...:/

For the first one, can you show that the median occurs at that point? If your distribution is symmetric, can you show the mean occurs at that point? (Hint: your distribution has 1/2 for a 0 value and 1/2 for a 1 value for every X_k).

I think i got it now...

and

I think i see the other 2 as well...