Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sorry if my question is trivial, it's been decades since I did any "real" math, so I'm quite rusty.

If I roll $N$ uniform $Y$-sided dice ($1\dots Y$), for the sum I get approximately a gaussian distribution that peaks at $A = \frac{N(Y+1)}2$.

How do I calculate the probability to get a sum of, say, $x A$ (where $1 < x < N\, Y/A$) ?

share|improve this question
1  
Perhaps you mean "the probability of getting a sum greater than $xA$" ? –  leonbloy Jan 25 '13 at 20:39

1 Answer 1

up vote 1 down vote accepted

Let $S_N$ denote the sum of $N$ results. Call $m_N=\max\{\mathbb P(S_N=k)\mid k\in\mathbb Z\}$, then $m_N$ behaves like $1/\sqrt{N}$ and, in particular, $\lim\limits_{N\to\infty}m_N=0$.

A quite different question is to estimate the behaviour or $q_N(x)=\mathbb P(S_N\geqslant xN)$ when $N\to\infty$, for some fixed real number $x$. Let $x^*=\frac12(Y+1)$. Then the following asymptotics hold. If $x\lt x^*$, $q_N(x)\to1$. If $x\gt x^*$, $q_N(x)\to0$. And $q_N(x^*)\to\frac12$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.