If we have a discrete random variable x and the data pertaining to it in X(n), how in matlab can we determine the probability mass function pmf(X)?
You can do this in at least eight different ways (some of them were already mentioned in the other solutions).
Say we have a sample from a discrete random variable:
X = randi([-9 9], [100 1]);
Consider these equivalent solutions (note that I don't assume anything about the range of possible values, just that they are integers):
[V,~,labels] = grp2idx(X);
mx = max(V);
%# TABULATE (internally uses HIST)
t = tabulate(V);
pmf1 = t(:, 3) ./ 100;
%# HIST (internally uses HISTC)
pmf2 = hist(V, mx)' ./ numel(V); %#'
%# HISTC
pmf3 = histc(V, 1:mx) ./ numel(V);
%# ACCUMARRAY
pmf4 = accumarray(V, 1) ./ numel(V);
%# SORT/FIND/DIFF
pmf5 = diff( find( [diff([0;sort(V)]) ; 1] ) ) ./ numel(V);
%# SORT/UNIQUE/DIFF
[~,idx] = unique( sort(V) );
pmf6 = diff([0;idx]) ./ numel(V);
%# ARRAYFUN
pmf7 = arrayfun(@(x) sum(V==x), 1:mx)' ./ numel(V); %#'
%# BSXFUN
pmf8 = sum( bsxfun(@eq, V, 1:mx) )' ./ numel(V); %#'
note that GRP2IDX was used to get indices starting at 1 corresponding to the entries of pmf
(the mapping is given by labels
). The result of the above is:
>> [labels pmf]
ans =
-9 0.03
-8 0.07
-7 0.04
-6 0.07
-5 0.03
-4 0.06
-3 0.05
-2 0.05
-1 0.06
0 0.05
1 0.04
2 0.07
3 0.03
4 0.09
5 0.08
6 0.02
7 0.03
8 0.08
9 0.05