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.

How to find the inverse function of the following: $f\left( x \right)=\sum_{i=0}^{x}{\frac{2^{i}}{i!}}$

share|improve this question
    
Why are you looking for an inverse? What do you need it for? It's good always to add some motivation, in order for others to help you in a better way. –  Andy Jan 4 '11 at 10:25
1  
Well, it requires the inverse of an incomplete gamma function... –  non-expert Jan 4 '11 at 12:03
    
Try to find a closed form. –  Raphael Jan 4 '11 at 12:27
    
Don't expect to find a closed form. –  leonbloy Jan 4 '11 at 16:23
    
I want to find the inverse CDF of the poisson distribution .. I found how to solve it somehow .. but still find that question interesting!! Can we find the inverse function of summations?? –  Osama Gamal Jan 6 '11 at 12:51

1 Answer 1

up vote 3 down vote accepted

In case you actually want to find the inverse cdf of the Poisson distribution (with parameter $\lambda=2$), there seem to be many statistical/mathematical softwares that can do the job for you. For example, see this.

EDIT: The following refers to the OP's question in a comment below (and above). Let $F$ and $F^{-1}$ denote a CDF and its inverse, respectively. Given $a>0$, define a function $F_a$ by $F_a (x) = aF(x)$. If $y=F_a (x)$, then $y=aF(x)$, hence, $F(x)=y/a$, and so $x=F^{-1}(y/a)$. It thus follows that $F_a^{ - 1} (y) = F^{ - 1} (y/a)$, where $F_a^{-1}$ denotes the inverse of $F_a$. Returning to the original question, take $F(x) = e^{ - \lambda } \sum\nolimits_{i = 0}^{\left\lfloor x \right\rfloor } {\frac{{\lambda ^i }}{{i!}}}$ and $a=e^\lambda$ (where $\lambda=2$). Then, $F_a (x) = \sum\nolimits_{i = 0}^{\left\lfloor x \right\rfloor } {\frac{{\lambda ^i }}{{i!}}}$. Hence we are done, by the relation $F_a^{ - 1} (y) = F^{ - 1} (y/a)$. Note: Here, $F$ is an increasing step function, hence not invertible in the usual sense. However, the inverse can be defined as described in the link above.

share|improve this answer
    
Of course, the factor of $e^{-\lambda}$ should be added. –  Shai Covo Jan 4 '11 at 13:16
    
Based on the OP's previous questions, this probabilistic interpretation of the question seems appropriate. –  Shai Covo Jan 4 '11 at 14:28
1  
Yeah, I was trying to find the inverse CDF of the Poisson distribution. I did find a way to do it .. But still want to know how to find the inverse function of the summation or is it impossible to find?? –  Osama Gamal Jan 6 '11 at 12:53

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.