Discretizing a continuous distribution

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alpedhuez
alpedhuez 2018-3-29
评论： Walter Roberson 2019-5-1
Suppose I have a continuous pdf on [0,1]. Suppose I have grid points {0,0.1,...,1}. Suppose I calculate f(0),f(0.1),...,f(1) and normalize by \sum{f(0)+....+f(1)}. Would it give an approximate discretized distribution?

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回答（2 个）

Xiaolu Zhu
Xiaolu Zhu 2018-12-29
Did you find the answer of how to discretize a continuous distribution?
  2 个评论
S.R.
S.R. 2019-5-1
I also would like to know if anyone knows how to discritize a continious distribution such as normal distribution or exponential distribution knowing its mean and standard deviation.
Walter Roberson
Walter Roberson 2019-5-1
Do you want equal spacing on the independent variable? Do you want to know where the boundaries are for equal spacing on the cdf? Do you want to divide up a range so that in each section the product of the pdf at the center point times the bin width is equal for all the bins?

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Walter Roberson
Walter Roberson 2018-12-29
No, it would not.
Consider the beta distribution with alpha=2, beta=5; https://en.wikipedia.org/wiki/Beta_distribution
x = 0:.1:1;
Alpha = 2; Beta = 5; %do not use beta, we need the beta function
num = x.^(Alpha-1).*(1-x).^(Beta-1);
den = gamma(Alpha)*gamma(Beta)/gamma(Alpha+Beta);
f = num./den;
fnorm = f./sum(f);
disp(mat2str(fnorm))
syms t;
Betareg = @(x,a,b) vpaintegral(t^(a-1).*(1-t)^(b-1),t,0,x)./beta(a,b);
fcdf = double(arrayfun(@(X) Betareg(X, Alpha, Beta), x));
disp(mat2str(fcdf))
[0 0.201845869866174 0.25202276572835 0.221596677434241 0.159483156437471 0.0961390555299185 0.0472542685740655 0.0174434702353484 0.00393785571450546 0.000276880479926165 0]
[0 0.114265 0.34464 0.579825 0.76672 0.890625 0.95904 0.989065 0.9984 0.999945 1]
Notice that f rises and falls but the cdf doesn't. So you might want cumsum(f)./sum(f) which would give you
[0 0.201845869866174 0.453868635594524 0.675465313028765 0.834948469466236 0.931087524996154 0.97834179357022 0.995785263805568 0.999723119520074 1 1]
You can see that the approximation is not good at all.

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