Poisson disribution calculation

I assume you mean that p is specified and you want to find a t that is mapped to it? The function p(t) = L*t*exp(-L*t) [L is your Lambda, exp(x) is your e^x] has two values of t mapped to each p in [0,1/e), but only one value mapped to p = 1/e. The derivative is p'(t) = L*(1-L*t)*exp(-L*t). It is zero when t = 1/L. The function is a maximum at p(1/L) = 1/e. You want to choose q and solve p(t) = q for the values of t. It is necessary that 0 <= q <= 1/e. If q = 1/e, the answer is immediate. The rest of this post assumes q < 1/e.

Use Newton's method applied to the function f(t) = p(t) - q. The initial guess is some value t[0]. The iterates are t[n+1] = t[n] - f(t[n])/f'(t[n]), for n >= 0. So you have t[n+1] = t[n] - (L*t[n]*exp(-L*t[n]) - q)/(L*(1-L*t)*exp(-L*t[n])).

Let r0 and r1 be the values for which p(r0) = q and p(r1) = q, with r0 < 1/e < r1. If you choose t[0] = 0, or any value smaller than r0, the numerator of the last term (the fraction term) in the Newton's iterate is negative since p(t) is an increasing function on [0,1/e]. The denominator of that fraction is positive, since 1-L*t > 0 for t in [0,1/e). That means t[1] = t[0] + positiveQuantity, and so t[1] > t[0]. This will happen on successive iterates, t[n+1] > t[n], due to the same sign-based argument and due to f"(t) < 0 on [0,1/e). You have an increasing sequence bounded above, so it has a limit. The Newton's iterates will converge to r0.

Since you want the *first* time (I think), there is no need to analyze the algorithm for obtaining r1.

Thank you very much for your help.