00:01
Hello everyone in this question it is given that x and y are random variables having the joint density function given by f xy and this is equal to 4xy for 0 less than equal to x less than x less than 1 and 0 otherwise and let g u be the joint density of the random variable u which is defined as x divided by y and another random variable v which is equal to y.
00:34
Now we need to find the marginal probability density function of gv.
00:43
Now in order to find the joint density of u and v, we use the relation g uv, that is the joint density of u and v is equal to f xy, the joint density of u and v, is equal to fxy, the joint density of x and y written in terms of u and v divided by the modulus of j here j is called the jacobian and it is defined by del u over del x that is the partial derivative of u with respect to x keeping y constant multiplied by del v over del y minus del v over del x multiplied by del v over del x multiplied by tell u over tell y.
01:31
Now using this relation let us find the joint density of u and v.
01:38
In the problem it is given that u is equal to x divided by y and v is equal to y.
01:54
Therefore, we can write del u over del x is equal to 1 divided by y and del u over del y is equal to x multiplied by y -d -to -the -power del y is equal to x multiplied by minus 1 divided by y -rest -to -the -power 2.
02:26
And in a similar fashion we can write del v over del x is equal to 0 and del v over del y is equal to 1.
02:49
Therefore using the formula for the jacobian we shall obtain the jacobian that is j as del u over del x multiplied by del v over del y minus del v over del x multiplied by delta v over del y minus del v over del x multiplied by del u over del y so this will give us del u over del x is 1 divided by y multiplied by del v over del y is 1 minus del v over del x is 0 multiplied by del u over del y is minus x divided by y rest to the power 2.
03:45
So our jacobian is equal to 1 divided by y and in terms of u and v since v is equal to y we can write it as 1 divided by v.
04:02
Therefore, we can write the joint density of u and v, that is g uv in terms of jacobian as f of xy now since u is x divided by y, therefore x is equal to u multiplied by y.
04:28
And since v is equal to y therefore x can be written as u multiplied by v comma y can be written as v and this is divided by the modulus of the circobian that is one divided by v so we can write this as v multiplied by the function u v comma v...