Question

An urn has 6 red and 4 white balls. Two balls are chosen at random and without replacement.
Let Y be the number of red balls among those selected.
a. Find the probability function (pmf) of Y.
b. Find the moment-generating function of Y.

Answer 1

As 2 balls are selected at random from 10 balls,

Let Y be the number of red balls among the two ball selected

Hence Probability No red balls are selected = Choosing 2 white balls = 4/10*3/9 = 2/15

Probability of 1 red ball selected = Choosing 1 red ball, 1 white ball = 4/10*6/9 + 6/10*4/9= 8/15

Probability of 2 red balls selected = Choosing 2 red balls = 6/10*5/9 = 5/15

Hence Probability function of Y is as follows

P(Y) = 2/15 when Y = 0

8/15 when Y = 1

5/15 when Y = 2

0 otherwise

B) Moment generating function of Y can be given by