Consider two independent random variables X1 ∼ U(0, 7), X2 ∼ U(0, 1). For the random variables Y = X1 + X2:

• What is the support of the distribution? Is the distribution continuous, discrete or mixed?

• Draw the probability distribution function of the random variable? Justify your answer with either theoretically or logically.

1. What is a discrete probability distribution? What are the two conditions that determine a probability distribution? What is a discrete probability distribution? Choose the correct answer below 

A. A discrete probability distribution exclusively lists probabilities O

B. A discrete probability distribution lists each possible value a random variable can assume. C. A discrete probability distribution lists each possible value a random variable can assume. together with its probability.

D. None of the above

2. What are the two conditions that determine a probability distribution? Choose the correct answer below

A. The probability of each value of the discrete random variable is between 0 and 1, inclusive, 

B. The probability of each value of the discrete random variable is between 0 and 1, inclusive.

C. The probability of each value of the discrete random variable is greater than 0 and less than

D. The probability of each value of the discrete random variable is greater than 0 and less than and the sum of all the probabilities is 1. and the sum of all the probabilities can be any amount. 1, and the sum of all the probabilities is 1. 1, and the sum of all the probabilities can be any amount.

You are given a fair die, i.e. the outcomes of any throw are contained in the sample space   Let the event A be that an even number is thrown and the event B that a multiple of three is thrown. Now consider the two random variables X and Y. X takes on a value of 1 if the event A has occurred and the value 0 if A has not occurred. The random variable Y takes on the value of 1 if the event B has occurred and the value zero if B has not occurred.

(a) Derive the joint pdf of X and Y. Provide this in the form of a table

                                        Y=0                                Y=1

b) What is the marginal probability distribution of Y?
(c) What is the conditional probability distribution of X, given that Y = 1?
(d) What is E (X|Y = 1)?
(e) What is Cov (X, Y )?
(f) Are X and Y independent random variables?