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.

Is there any known polynomial bound of the Erlang distribution? I'd like to say that, given $k$ and $\lambda$ with probability p the r.v. is going to be less than some value x.

share|improve this question
    
Let $X_k \sim \mathrm{Erlang}(k,\lambda)$. Then $\mathbb P(X_k \leq t) = \mathbb P(N(t,\lambda) \geq k)$ where $N(t,\lambda) \sim \mathrm{Pois}(\lambda t)$. Now you can use your favorite tail bound on the Poisson distribution. –  cardinal Nov 10 '11 at 23:27

1 Answer 1

up vote 1 down vote accepted

That is simply the cumulative distribution function, given in WP by $\gamma(k,k\lambda)/(k-1)! = 1-\sum_{n=0}^{k-1}\mathrm e^{-\lambda x}(\lambda x)^{n}/n! $, where $\gamma$ is the incomplete gamma function.

share|improve this answer
    
I know, but I wanted to see if there is an approximation that does not involve factorials –  ACAC Apr 21 '11 at 20:32
    
Why that constraint? If $k$ is large, you can use Stirling approximation. There are probably approximations for the incomplete gamma function too. –  Emre Apr 21 '11 at 20:36

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.