# Probability Homework

Probability Homework

A soft drink machine has a random variable X in supply at the beginning of a given

day and dispenses a
random amount Y during the day (with measurements in gallons). It is not resupplied

during the day, so we know
that Y = X. It has been observed that X and Y have the following joint density:
f(x,y) = ½ 0 = y = x = 2
a. Find the conditional distribution of Y given X = x. Verify that f(y|x) is a

valid density function.
b. What is the probability that less than 0.5 gallons are sold, given that the

machine contains 1.5 gallons at
the start of the day?
c. Find the expected sales given that 1.5 gallons are supplied at the start of the

day.
d. Find the standard deviation of sales given that 1.5 gallons are supplied at the

start of the day.
e. Find the Cov(X,Y).
f. Compute the correlation coefficient. What does it tell you?

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