/sci/ - Science & Math

Hi, Clint Eastwood here. So, all this "cosmic inflation" stuff. They thought this up to explain why the universe is smooth and not lumpy, right? My question is... why is that something that needs explaining in the first place? Why couldn't the universe simply have been really smooth from the beginning?

n a natural number, 0 < p < 1, and d a real such that

(n choose d)*p^(d choose 2) = 1.

Show that, if k ≥ d + 1, then as n tends to infinity,

(n choose k)*p^(k choose 2)

tends to 0.

Any ideas how to do this?

0 < p < 1. n some natural number.

d is a real number chosen such that (n choose d)*p^(d choose 2) = 1.

I need to prove that if k is greater than or equal to d + 1,

(n choose k)*p^(k choose 2), i.e. the same thing but with d replaced by k, tends to 0 as n tends to infinity.

Wut do? What's the general approach?

n independent trials, X1, ... , Xn, with probability p. The expected number of successes, X = X1 + ... + Xn, is np.

Apparently, the probability of more than np/2 successes is greater than a half, i.e. P(Xn > np/2) > 1/2.

Why? :/