00:01
In this problem, we are given the following joint density function f of x y as c times x times 1 plus y when x is between 0 and 1 and y is between 0 and 6 and 0 otherwise.
00:21
And we have three parts.
00:24
First, what is the value of this constant c? so this function should be normalized over the entire xy plane or in the relevant part.
00:40
So we have integral from 0 to 1 dx, integral from 0 to 6 dy, c times x times 1 plus y.
00:50
This should be equal to 1.
00:52
We will have all these polynomials in the integrals, so these are elementary functions and therefore let me just write down the integrated results.
01:03
The y integral returns 24 times c times x and with that the x integral gives 12c.
01:14
So we see that c is equal to 1 over 12.
01:19
Next, we are going to compute the probability that x is less than or equal to 1 and y is less than or equal to 1.
01:31
So we impose the following cuts x less than or equal to 1, y is less than or equal to 1, dx, dy, f.
01:44
So in this relevant part namely where x is between 0 and 1 and y is between 0 and 6, our limits become from 0 to 1 for dx and from 0 to 1 for dy.
01:58
And we have 1 over 12 x times 1 plus y.
02:03
So again let me just write down the results for these elementary integrals.
02:07
The y integral gives x over 8 and the x integral gives 1 over 16...