Question

Among a group of ten people, two are NDP, three are Liberal and four are Conservative. Three of the eight are selected at random, without replacement. Find the joint pmf of the random vector (X; Y ), where X denotes the number of NDP selected and Y denotes the number of Liberals selected. Be sure to specify its range.
The final answer is supposed to be: ((2choose x)(3choose y)(4 choose 3-x-y))/(10 choose 3)

please show the steps and necessary explanations of getting the above final result.

Answer 1

Here by the problem,

Among a group of 10 people, 2 are NDP, 3 are Liberal and 4 are Conservative. 3 people are selected at random, without replacement.

Now as the choice has been done by without replacement, hence out of 10 people 3 people can be chosen (hence the total number of possible choices) in ways.

Now the joint pmf of the random vector (X; Y ), where X denotes the number of NDP selected and Y denotes the number of Liberals selected be as follows,

Note that the possible values of X be x=0,1,2 as there are 3 NDP

On the other hand the possible values of Y be y=0,1,2,3 as there are 3 liberal.

Now as there are 3 people in total selected hence x+y should be less than or equal to 3 since there other possibilities (like from 4 conservatives etc)

So the pmf of (X,Y) be,

=P(x NDP are chosen from 2 NDP, y Liberal are chosen from 3 Liberal and the rest 3-x-y are chosen from rest 10-2-3=5 other people)

Now x NDP are chosen from 2 NDP in ways, for each of such ways y Liberal are chosen from 3 Liberal in ways and the rest 3-x-y are chosen from rest 5 other people in ways.

Hence the total number of ways be in favourable of such choice,

where

So the required pmf be

where

Hence the answer..............

Thank you.................