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.

a. (3 points) Draw a picture of the region in the X-Y plane for which the joint density of X and Y is

non-zero.

b. (3 points) Compute the joint density function for X and Y . (As always, make sure you write a complete

expression.)

c. (3 points) Compute the marginal probability density for Y , fY (y).

d. (3 points) Compute the conditional probability density of X, conditional on Y = y, fX|Y (x|y). (Make

sure you state the values of y for which this exists.)

Pay extra attention for part d, because it might be very tricky. I am not sure if it is 1/2 or a ln function? Please help me out.