Theoretical Plot of Probability Density Function

Question

I'm using the plot the probability density function:

y = zeros(1,10000);
for j=1:10000
    r = rand(100,1);
    for i=1:100
        y(j) = y(j) + r(i) - 0.5;
    end
    y(j) = y(j)/sqrt(100);
end
[n,x] = hist(y,100);
plot(x,n/10000/diff(x(1:2)));
hold on;

However I'd also like to print the theoretical too. The best I seem to have managed is the following:

plot(x,normpdf(x,0,1),'r');

But this doesn't follow the actual at all. What am I missing here? Here's what my plots look like at the moment. The blue is the actual and the red is the theoretical.

enter image description here

Your theoretical sigma is clearly wrong, try for example plot(x,normpdf(x,0,1/pi),'r'); — Maksim Gorkiy
@ClockworkOrkwork that worked. Of course my sigma was wrong...thanks! :) I can't quite accept your comment as an answer though... ;) — codedude

kishkash kishkash · Accepted Answer · 2014-10-16T20:44:38

Your y's aren't coming from a uniform distribution; They're coming from a distribution of a sum of i.i.d (independent identically distributed) random variables of uniform distribution with mean 0 and variance 1/12. The sum approaches normal distribution as the number of summed variables (100 in your case) gets large. Using your code, I was able to achieve a very nice fit with normpdf, with the correct variance being 1/12 (sigma is the square root of this number):

y2=normpdf(x,0,sqrt(1/12));
plot(x,y2,'r');

BTW, your matlab code can be made simpler and more readable replacing the first 8 lines with:

r=rand(100,10000)-0.5;
y=sum(r)/sqrt(100);

Theoretical Plot of Probability Density Function

1 Answers