Product of two uniform random variables on the interval [0,1]

[Absolut_Ideal]

Homework Statement

If R1 and R2 are two uniformly distributed random variables on the interval [0,1]. What is the density function Z=R1*R2?

Homework Equations

I'm not sure actually

The Attempt at a Solution

I have tried to manipulate with moment generating function (which i suspect is the method i should use) but I have not come up with anything good.

I know that since they are independent the Expected Value of Z should be 1/2*1/2=1/4. But I'm not sure if this will help me at all.

I'm really just looking for an answer to whether I should continue experimenting with moment generating function or if I should try a different approach to the problem. Currently, I have no idea where to even start. The book for the course does not provide any examples what so ever for these types of problems.

Thanks in advance!

 

[Simon Bridge]

When you add two distributions, the resulting distribution is their convolution.
Multiplying is the same as adding the logarithms, so...

 

[Ray Vickson]

A standard method to get the distribution of h(X,Y) for independent X and Y is to first get the cdf P{h(X,Y) <= w}, which you can get as
$$P\{ h(X,Y) \leq w \} = \int_{-\infty}^{\infty} P\{h(X,Y) \leq w | Y=y\} f_Y(y) \, dy,$$ and to recognize that $P\{h(X,Y) \leq w | Y=y\} = P\{h(X,y) \leq w \},$ because X and Y are independent.

RGV

 

[Simon Bridge]

Oooh - I'd actually forgotten about that. Probably since all I've every had to do is either add or multiply distributions. Mostly add. Thanks.

 

[ehild]

Homework Statement

If R1 and R2 are two uniformly distributed random variables on the interval [0,1]. What is the density function Z=R1*R2?

Follow Ray Vickson's hint and determine the cumulative distribution function first: If R1 takes the values x, R2 takes the values y find the probability that z=xy < w: F(w)= P(z<w)

R1 and R2 are uniformly distributed. The (x,y) ordered pairs can be represented as points of the xy plane in the range 0<x<1, 0<y<1, and the probability of finding a point in a given domain is proportional to the geometric area. Draw the curve xy=w, and find the area where 0<x<1, 0<y<1, and xy<w (painted yellow in the picture).

ehild

 

[Simon Bridge]

Hmmmm... not told that X and Y are independent, or the range on which they are uniform.
Only need the density fn not the cumulative density.

What's wrong with taking the antilog of the convolution of the logs?
I think we now need to hear from OP.

 

[Ray Vickson]

Hmmmm... not told that X and Y are independent, or the range on which they are uniform.
Only need the density fn not the cumulative density.

What's wrong with taking the antilog of the convolution of the logs?
I think we now need to hear from OP.

We were told they are uniform on [0,1], but not that they are independent. Even though we want the density, sometimes the easiest way to get that is to differentiate the cumulative.

RGV

 

[Simon Bridge]

We were told they are uniform on [0,1],

Gah! read that twice and missed it both times ... right: time to stop for the night!

 

[Absolut_Ideal]

Thanks a lot for the explanations guys! It helped me out a lot!

Cheers!

 

[ehild]

Thanks a lot for the explanations guys! It helped me out a lot!

Cheers!

What have you got for the density function f(Z)?

ehild