Solving a Probability Question with Poisson Distribution

persist911
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Homework Statement


I am solving a particular probability question using Poission distributin after the Solving I get an equation

Homework Equations




Code:
 1.67 =  e^-a (1/1-a)
I ought to get the value of a from the equation but I was Unable to go further from here .

The Attempt at a Solution



I have tried expaning e^-a using series but no luck . I am totally stuck.
 
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Assuming you meant
1.67 = \frac{e^{-a}}{1 - a}
there is no "nice" solution.
It can be written as
1 - a = W_n(- a / e) \qquad (n \in \mathbb{Z})
where Wn the so-called product-log function, i.e. the solution for w of z = w ew. So that doesn't bring you much further, I'm afraid... are you sure you need to solve this analytically and - if so - the equation is correct?
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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