What is the pdf of multiplication of two random variables?

[iVenky]

We have two independent random variables X and Y whose pdfs are given as f(x) and f(y). Now when you multiply X and Y you get a random variable say Z. Now what is the resulting pdf f(z)?

I mean how is that related to the pdf of f(x) and f(y)?

From what I read it looks like

f(z)=f(x) * f(y)

where "*" represents convolution.

But I couldn't find how you get that.
Thanks a lot :)

 

[mfb]

Convolution is the operation for the sum of two variables.

You can get f_z via a double integral
$$P(z<Z)=\iint_{x'y'=z} dx' dy' f_x(x') f_y(y')$$ where the integration limit is a hyperbola.
You can split this into two parts with explicit integration limits like that:

$$P(z<Z)=\int_{-\infty}^0 dx' f_x(x') \int_{z/x'}^\infty dy' f_y(y') + \int_0^\infty dx' f_x(x') \int_{-\infty}^{z/x'} dy' f_y(y')$$

f_z is the derivative of that.

 

[iVenky]

Convolution is the operation for the sum of two variables.

You can get f_z via a double integral
$$P(z<Z)=\iint_{x'y'=z} dx' dy' f_x(x') f_y(y')$$ where the integration limit is a hyperbola.
You can split this into two parts with explicit integration limits like that:

$$P(z<Z)=\int_{-\infty}^0 dx' f_x(x') \int_{z/x'}^\infty dy' f_y(y') + \int_0^\infty dx' f_x(x') \int_{-\infty}^{z/x'} dy' f_y(y')$$

f_z is the derivative of that.

I know the meaning of convolution but what I would like to know is how multiplication of 2 random variables results in a pdf which is the convolution of the two pdfs.

That's what I would like to know.

Thanks a lot :)

 

[mfb]

It does not.
But I gave one way to calculate the function in my post. You can even derive the whole equation (careful with the limits) to get a more direct expression for f(z).