Question

Let X be the minimum and Y the maximum of three numbers drawn, without repositioning, from the set {0, 1, 2, 3, 4}.

a) Determine the joint probability function of X and Y.

b) Calculate the marginal probability functions of X and Y.

c) Are X and Y independent? Justify.

d) Find the probability function of Z = Y - X

Answer 1

(a)

Three numbers are drawn without replacement from . So, we obtain when two numbers drawn are x and y and third one is an integer in open interval .

Further number of all possible draws is .

Thus joint probability mass function is given by

Corresponding joint probability mass function is as follows.

(b)

Marginal probability mass function of X is as follows.

Marginal probability mass function of Y is as follows.

(c)

Two random variables X and Y are independent if for all possible values of X and Y.

Here for X=2, Y=2 (we might consider any other such one as only one contradiction serves our purpose) we observe that

So, X and Y are not independent.

Note- In fact, from our construction of joint probability mass function we observe that there is exact condition regarding random variables as which suggests dependency of random variables.

(d)

Using contingency table denoting joint probabilities of X and Y we get distribution of as follows.