Find intersection of 2 normal distribution

Hi,

I have 2 normal pdf and I want to find their intersection:

the intersection is between 160 to 170.

I have code as follows:

mu1 = 160;  
var1 = 20;
mu2 = 175;
var2 = 15;
yfun = @(mu,var, x)(2*pi*(var))^(-0.5)* exp(-((x-mu).^2)/(2*(var)));
val = fzero(@(x) yfun(mu1, var1, x) == yfun(mu2, var2, x), rand * (mu1 - mu2) + (mu1 + mu2))

Output:
val =
  324.6802

Its the value of 2nd parameter of fzero() and fzero() sets val to it's 2nd parameter whatever i change it to.

How do i find the intersection's x value?

Answer 1

[Star Strider]

The ‘val’ value is the x-value of the intersection, however you need to start fzero in the correct region for it to return the correct value. It is then straightforward to calculate the y-value from either Gaussian function, since they are both approximately the same at that point.

I changed your code slightly so it returns the correct results:

mu1 = 160;  
var1 = 20;
mu2 = 175;
var2 = 15;
yfun = @(mu,var, x)(2*pi*(var))^(-0.5)* exp(-((x-mu).^2)/(2*(var)));
val = fzero(@(x) yfun(mu1, var1, x) - yfun(mu2, var2, x), mean([mu1,mu2]))
yval = yfun(mu1, var1, val)

producing:

val =
          167.872647061767

yval =
        0.0189439821229373

Experiment to get the result you want.

[Aishwarya Radhakrishnan]

Thanks ! it worked

[John D'Errico]

Note the importance of how Star used subtraction there instead of using ==. Testing for equality is a bad idea, because it will essentially never happen that they are exactly equal. You want to search for where the difference crosses zero.

[Star Strider]

@Aishwarya Radhakrishnan — As always, my pleasure!

@John D’Errico — I very much appreciate your Comment!