# How to estimate probabilities of an arbitrary range, based on the probability distribution of a given a data set of numbers?

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Clarisha Nijman on 22 Oct 2018
Commented: Clarisha Nijman on 23 Oct 2018
Hello,
Given a series of values x, I want to estimate the probabilities of a range of numbers U, in(using) the probability distribution of the given series x. My code works for one value, but I need probabilities of a range, Can somebody give me some feedback please?
This is the code:
%%Generate some data/series
x=randi([-2 50],25,1);
%Values/ranges of interest
U=[-100:100];
%define histogram and probability distribution of x
h = histogram(x);
h.Normalization = 'probability';%Changing count in probabilities
h.Values(U); %finding probabilities of range U

Bruno Luong on 22 Oct 2018
Edited: Bruno Luong on 22 Oct 2018
Use HISTCOUNTS then
N = histcounts(x, [-Inf, U, Inf]);
P = N(2:end) / sum(N)

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Bruno Luong on 22 Oct 2018
No that is not problem. There is 200 intervals from 201 edges.
• Interval 1, [-100,-99)
• Interval 2, [-99,-98)
• ...
• Interval 200, [99,100)
However if you want to include the "tail" [100,Inf) as well then go to the last.
N = N(2:end) / sum(N)
Just wonder why you keep the tail but not the head.
For such detail, may be you ought to read the doc of HISTCOUNTS carefully and adapt the code to your need rather than take my code literally.
Clarisha Nijman on 22 Oct 2018
Ok, that is a good idea to study this topic again in Matlab, with this new insight you gave me today!
Thank a lot!
Clarisha Nijman on 23 Oct 2018
x=randi([-3 3],10,1); U=[-5:5];
N = histcounts(x, [-Inf, U, Inf ]) prob = N(2:end) / sum(N)
%alternative code f=hist(x,U); prob=f/sum(f);
Now I fully understand your answer. With this small example it is clear. With the tails you are getting 2 extra intervals. An arbitrary value for U, let's say 2 is associated with interval <1,2] Such that we have eleven intervals, and since the left tail does not live in U, it is excluded, and that's why use (2:end) in the code. Thanks a lot!

Torsten on 22 Oct 2018
%%Generate some data/series
X=randi([-2 50],25,1);
%Values/ranges of interest
U=[-100:100];
X = sort(X)
[countsX, binsX] = hist(X)
cdfX = cumsum(countsX) / sum(countsX)
extrap_left = (min(U) > max(X));
extrap_right = (max(U) > max(X));
p_U_left = interp1(binsX,cdfX,min(U),'linear',extrap_left)
p_U_right = interp1(binsX,cdfX,max(U),'linear',extrap_right)
p_U = p_U_right - p_U_left

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Torsten on 22 Oct 2018
- Sort X.
- Count the number of occurences of each distinct element in X and divide by the number of elements of X. This gives you the empirical probability of the elements in X.
- For each u in U, look whether it is also an element of X. If no, assign probability 0, if yes, assign the empirical probability of u as an element in X.
Clarisha Nijman on 22 Oct 2018
If you want to use data you can not do that, that would be excluding situations that possibly might occur. That is why the frequency polygon is a smooth line. To estimate values in between.
Torsten on 22 Oct 2018
If you get discrete values from a random variable, say [ 1 2 4 5 6 ], how should it be possible to tell p({3}) ? (Hint: It's impossible).
In my opinion, the most reasonable estimate would be p=0 since it does not appear in the list.
If you know the distribution the values stem from, you can get a Maximum Likelihood Estimate (MLE) of the parameters describing the distribution. Having calculated these parameters, you can give estimates of probabilities for elements of your choice.

Bruno Luong on 22 Oct 2018
Edited: Bruno Luong on 22 Oct 2018
not sure, is it what you want?
x=randi([-2 50],10000,1);
U=[-100:100];
h = histogram(x, U);

#### 1 Comment

Clarisha Nijman on 22 Oct 2018
Let's say x is the profit of a shop observed 20 times. and the values are: 2,5,7,2,20,25,35,15,6,-2,15,27,2,20,15,5,7,2,20,25
This can be associated with a probability distribution. And you can plot it.
Now it is asked to estimate the probability of the values in between, and also in the tails. U=-[5 -4 -3 -2 -1 0 1 2 .... 40]