Integration of probability density function

Question

I would like to integrate the function so that I know the probability between values between 0 and 90000

MixtureDistribution[{0.43999526215889906`, 
   0.5600047378411006`}, {NormalDistribution[58745.18201580311`, 
    15134.735419294597`], 
   NormalDistribution[78262.1217564796`, 2797.225012305396`]}];

Answer 1

So given distribution:

dis=MixtureDistribution[
{0.43999526215889906`,0.5600047378411006`},
{NormalDistribution[58745.18201580311`,15134.735419294597`],
NormalDistribution[78262.1217564796`,2797.225012305396`]}];

with the PDF:

Plot[PDF[dis, x], {x, 0, 10^5}, PlotRange -> All,PlotTheme -> "Detailed"]

You can compute probability via built in function:

In[]:= Probability[0<=x<=90000,x\[Distributed]dis]
Out[]= 0.991409

which is the same as:

In[]:= NIntegrate[PDF[dis,x],{x,0,90000}]
Out[]= 0.991409

Answer 2

This is a quite standard way.

a[t_?NumericQ] :=  CDF[MixtureDistribution[{0.43999526215889906`, 
0.5600047378411006`}, {NormalDistribution[58745.18201580311`, 
 15134.735419294597`], NormalDistribution[78262.1217564796`, 2797.225012305396`]}], t]
a[90000] - a[0]

0.991409