Question

A fair coin is tossed three times independently: let X denote the number of heads 

on the first toss and Y denote the total number of heads. Find the joint probability 

mass function of X and Y .

a) Provide the joint distribution table of the above experiment.

b) Find the 

i) the marginal probabilities of X and Y

ii) P(X<Y)

Answer 1

(a) The joint distribution table is given as-

(b)

(i) The marginal probability

P(X) is given by -

P(Y) is given by-

(ii) $P(X<Y)=P(0,1)+P(0,2)+P(0,3)+P(1,2)+P(1,3)$

$=\dfrac{1}{4}+\dfrac{1}{8}+0+\dfrac{1}{4}+\dfrac{1}{8}=\dfrac{3}{4}$