Use the following result: if and are two independent random variables which have a density, namely for and for , then has a density , which is given by .
Let X and Y be independent exponential random variables with and
Find the joint probability distribution .
I know that the probability density function for X is for x>0 and 0 otherwise.
also
I know that the probability density function for Y is for x>0 and 0 otherwise.
I am not sure about jointly. Do I simply multiply them together?
Find .
I have no idea how to begin this part, although I know the answer is .04.
Can anyone help?
About the second question we have two independent exponential random variables X and Y ande their p.d.f are...
(1)
(2)
...where is the so called 'Heaviside step function' ...
Now if we set the p.d.f. of Z is given by the convolution of and ...
(4)
Using a basic property of the Laplace Transform we have that...
(5)
... so that...
(6)
(7)
(8)
... and from (8) taking the inverse Laplace Tranform...
(9)
Once You have the p.d.f. of Z You can obtain the requested probability by integration...
(9)
Kind regards