If Y1 is a continuous random variable with a uniform distribution of (0,1)
And Y2 is a continuous random variable with a uniform distribution of (0,Y1)
Find the joint distribution density function of the two variables.
Obviously, we know the marginal density functions of each variable. I am used to solving for the marginal density functions from the joint density functions but not the other way around. Since, the variables are not independent, I can't simply multiply the marginal density functions together. Any ideas?
To find the joint distribution density function of Y1 and Y2, we need to consider the relationship between the two variables. In this case, Y2 is defined as a continuous random variable with a uniform distribution between 0 and Y1.
We need to find the probability density function (PDF) f(Y1, Y2) that describes the joint distribution of Y1 and Y2. To do this, we can use the concept of conditional probability.
First, let's consider the range of Y1 and Y2. Since Y1 is defined to have a uniform distribution between 0 and 1, its PDF is given by:
f(Y1) = 1 for 0 <= Y1 <= 1
0 otherwise
Now, for each fixed value of Y1, we can determine the distribution of Y2. Since Y2 is defined to have a uniform distribution between 0 and Y1, its PDF given Y1 can be written as:
f(Y2|Y1) = 1/Y1 for 0 <= Y2 <= Y1
0 otherwise
Now, to find the joint distribution density function f(Y1, Y2), we multiply the PDFs of Y1 and Y2 given Y1:
f(Y1, Y2) = f(Y1) * f(Y2|Y1)
Substituting the expressions for f(Y1) and f(Y2|Y1) into the equation:
f(Y1, Y2) = 1 * (1/Y1) for 0 <= Y1 <= 1 and 0 <= Y2 <= Y1
0 otherwise
So, the joint distribution density function of Y1 and Y2 is given by:
f(Y1, Y2) = 1/Y1 for 0 <= Y1 <= 1 and 0 <= Y2 <= Y1
0 otherwise