1. ## Joint probability distribuction

I am not quite sure about how to solve the following exercise:
Given that a machine is turned on at a random time (X) (in hours) in a particular day and turned off at another random time (Y) (on the same day) determine the joint probability mass function f(X,Y).

Well I am certain of a thing! the the X marginal prob. mass function is a uniform one, as to the Y one I am not quite sure of it since y must always be greater than x.

Well even if I know the Y and X distribution, can I assume they are Independent? well the range of Y depends on x doesnt it make one dependent on the other?

2. You know the marginal distribution of X and the conditional distribution of Y. Multiply the corresponding pdfs to get the answer.

Strictly speaking, though, it bothers me that they don't specify that by "at a random time" they mean uniformly distributed.

3. Wth Is the conditional distribution of y? Never heard of it! How can I have it.? I only oboé that if I have te two marginal distributions and the variables are independent one of another qe can multiply them to get the houve distribution

4. X and Y are clearly not independent since the possible values of Y depends on X. The idea seems to be that X ~ uniform(0, 24) and that, given X, Y ~ uniform(X, 24).

You could try calculating the cdf explicitly and taking the mixed derivative to get the pdf if you don't know about conditional distributions yet. Calculate

$\displaystyle P(X \le x, Y \le y) = P(Y \le y | X \le x) P(X \le x)$.

Certainly, you must know about at least conditional probabilities.