Truncating Poisson distribution on desired support in Matlab

Question

I want to construct a 3-dimensional Poisson distribution in Matlab with lambda parameters [0.4, 0.2, 0.6] and I want to truncate it to have support in [0;1;2;3;4;5]. The 3 components are independent.

This is what I do

clear
n=3; %number components of the distribution 

supp_marginal=0:1:5;
suppsize_marginal=size(supp_marginal,2); 

supp_temp=repmat(supp_marginal.',1,n); 
supp_temp_cell=num2cell(supp_temp,1); 
output_temp_cell=cell(1,n);
[output_temp_cell{:}] = ndgrid(supp_temp_cell{:});
supp=zeros(suppsize_marginal^n,n);

for h=1:n
    temp=output_temp_cell{h};
    supp(:,h)=temp(:);  
end
suppsize=size(supp,1); 

lambda_1=0.4;
lambda_2=0.2;
lambda_3=0.6;

pr_mass=zeros(suppsize,1);
for j=1:suppsize
    pr_mass(j)=(poisspdf(supp(j,1),lambda_1).*...
                poisspdf(supp(j,2),lambda_2).*...
                poisspdf(supp(j,3),lambda_3))/...
                sum(poisspdf(supp(:,1),lambda_1).*...
                    poisspdf(supp(:,2),lambda_2).*...
                    poisspdf(supp(j,3),lambda_3));
end 

When I compute the mean of the obtained distribution, I get lambda_1 and lambda_2 but not lambda_3.

lambda_empirical=sum(supp.*repmat(pr_mass,1,3)); 

Question: why I do not get lambda_3?

Answer 1

tl;dr: Truncation changes the distribution so different means are expected.

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This is expected as truncation itself has changed the distribution and certainly adjusts the mean. You can see this from the experiment below. Notice that for your chosen parameters, this just starts to become noticable around lambda = 0.6.

Similar to the wiki page, this illustrates the difference between E[X] (expectation of X without truncation; fancy word for mean) and E[ X | LB ≤ X ≤ UB] (expectation of X given it is on interval [LB,UB]). This conditional expectation implies a different distribution than the unconditional distribution of X (~Poisson(lambda)).

% MATLAB R2018b
% Setup
LB = 0;   % lowerbound 
UB = 5;   % upperbound

% Simple test to compare theoretical means with and without truncation
TestLam = 0.2:0.01:1.5;
Gap = zeros(size(TestLam(:)));
for jj = 1:length(TestLam)
    TrueMean = mean(makedist('Poisson','Lambda',TestLam(jj)));
    TruncatedMean = mean(truncate(makedist('Poisson','Lambda',TestLam(jj)),LB,UB));
    Gap(jj) = TrueMean-TruncatedMean;
end

plot(TestLam,Gap)

Notice the gap with these truncation bounds and a lambda of 0.6 is still small and is negligible as lambda approaches zero.

lam = 0.6;    %  <---- try different values   (must be greater than 0)
pd = makedist('Poisson','Lambda',lam)
pdt = truncate(pd,LB,UB)
mean(pd)                         % 0.6
mean(pdt)                        % 0.5998

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Other Resources:
1. Wiki for Truncated Distributions
2. What is a Truncated Distribution
3. MATLAB documentation for truncate(), makedist()
4. MATLAB: Working with Probability Distribution (Objects)