How to normalize a probability density function?

When I run this code, the pdf curve does not line up with the normalized histogram. How can I fix this so that the area under the curve is 1?

rng(0); %initialize seed

x=randn(1000,1); %create matrix of random numbers

width=8/29; %size of bin, uses 29 so there will be 30 total bins

binranges=-4:width:4; %sets range of values

[bincounts]=histc(x,binranges); %counts how many x are in each bin

[norbincts]=bincounts./1000; %normalize so area=1

figure %start image

bar(binranges,norbincts,'histc') %add bars

axis([-4 4 0 0.5]) %resize the axes

hold on %keep same plot

M=mean(x); %mean of matrix

S=std(x); %standard deviation of matrix

gauss=pdf('normal',binranges,M,S); %create curve of gaussian dist

plot(binranges,gauss) %add gauss dist to plot

text='Mean = %f Standard deviation = %f'; %title

str=sprintf(text,M,S); %title

title(str); %title

Answer 1

[Amos]

As I understand it, the Gauss curve is normalized:

trapz(binranges,gauss),trapz(binranges,norbincts)

leads to an output

ans =

      0.9999

ans =

      0.2759

It seems that the curve is normalized in the sense that the integral is 1, whereas the histogram is normalized in the sense that the sum is 1 (which makes a difference, as the integral is the sum times the bin width).

In order to get the curve over the bars, you can either multiply the curve by width or divide the bars by width.