area under a pdf not 1?

I have convoluted two pdfs. Both of my pdfs are exponential with lambda=10 and 20, respectively. I have used the following code:

    fun1=@(x)exppdf(x,10);
    fun2=@(x)exppdf(x,20);
    x=0:30;
    y=conv(fun(x),fun2(x));

The code above gives the vector y with 61 elements in it. When I used the 'trapz(0:60,y)' function to find the area under the curve y, it gives 0.80 as output. However, the area should be one since y represents a pdf. I suspect that it is due to the error associated with integration via 'trapz'.

But I need to confirm that I am getting a valid pdf. What function can I use instead of trapz. I cannot use the function 'integral', because it requires functional form of convolution output which I don't have.

Any suggestions?

Answer 1

[Star Strider]

It has approximately unity area:

fun1 = @(x)exppdf(x,10);
fun2 = @(x)exppdf(x,20);
y = @(x) conv(fun1(x),fun2(x));

x = linspace(0, 100);
yint = trapz(y(x))

yint =
       1.0474

It is necessary to bear in mind the support of PDFs. Integrate it over its entire length.

[Abhinav]

Oh, I get it. Thanks!

[Star Strider]

My pleasure!

[Abhinav]

Hi, when I replace 10 and 20 by 5 and 3, respectively (keeping everything else same), I get area equal to 1.25. Any idea why it is happening?

[Star Strider]

Not really. After experimenting a bit, it seems to be a function of the length of ‘x’, so there must be some sort of normalization factor involved. Beyond that, I’m not sure. I’ll think about this and if I come up with anything, I’ll post back.

[Abhinav]

Okay.