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functionderivativemaplepoisson

The positive real number at which is the derivative of the Poisson density equal to 0

I want to find the extreme of the Poisson density function as a smooth curve of the Poisson density. For that I need to obtain the unique positive numerical value where is the derivative of the Poisson density equal to 0.

With this:enter image description here I get only the number -0.4276 which is not >0.

The following plots nicely the appropriate curves but do not give the numerical extreme. Here, x is to be thought as the parameter $\lambda$. This extreme value should be approximately as big as $x$ which is E(X) but not exactly the same. enter image description here

Solution

  • You didn't inform fsolve that you wanted a positive root.

    restart;
ee := add(k*x^k*exp(-x)/k!,x=2..2):
dee := diff(ee,k):

fsolve(dee,k=0..10);

      2.479687450

plot({ee,dee}, k=0..10);

    enter image description here

    This also works here:

    fsolve(dee,k=0..infinity);

      2.479687450