Question

You have a ruler of length 2 and you choose a place to break it using a uniform probability distribution. Let random variable X represent the length of the left piece of the ruler. X is distributed uniformly in [0,2] . You take the left piece of the ruler and once again choose a place to break it using a uniform probability distribution. Let random variable Y be the length of the left piece from the second break. Draw a picture of the region in the X-Y plane for which the joint density of X and Y is non-zero.

Answer 1

Hi!

When you break the rule for the first time, de probability distribution of X is constant in [0, 2], and is zero outside. If the the value obtained is X = a, then the probability distribution of Y (conditioned on X = a) is constaint in interval [0, a], and zero outside. Then the region in the X-Y plane where the joint density is non-zero is the filled triangle  in the figure.