Question

In a hospital there are 9 male patients and 6 female patients and tested positive for coronavirus and they need ventilatos.But in hospital there are only 5 ventilators available. We selects 5 patients at random for ventilators.

1) If M represents the numbers of male patients and F represents the numbers of female patients then find the joint PMF of M and F. Write the matrix form of joint pmf.

2) P_M|F(m|f=2)?

3) Prove the following relation, V ar(M|F) = E(M²|F) - (E(M|F))²

4) Prove the dependency or independency of these two random variables.

Answer 1

a) The joint PMF for the number of males and females selected for ventilators is computed here as:

p(m, f) = Number of ways to select m males from 9 males * Number of ways to select f females from 6 females, Total ways to select 5 people from 15 people

Using the above formula, we get the matrix form of the joint PMF here as:

	m = 0	m = 1	m = 2	m = 3	m = 4	m = 5
f = 0	0	0	0	0	0	0.04195804
f = 1	0	0	0	0	0.25174825	0
f = 2	0	0	0	0.41958042	0	0
f = 3	0	0	0.23976024	0	0	0
f = 4	0	0.04495504	0	0	0	0
f = 5	0.001998	0	0	0	0	0

b) For f = 2, we know that there is only 1 value of m possible.

P(m = 3 | f = 2) = 1
P(m = i | f = 2) for all other values of m = i.

c) E(M | F = 2) = 3*P(m = 3 | f = 2) = 3
E(M2 |F = 2 ) = 32*P(m = 3 | f = 2) = 9

Now we know here that the variance is 0 as there is only 1 possible value here, therefore:
Var(m | F = 2) = 0

Now, E(M2 | F = 2) - [E(M | F = 2)]2 = 9 - 32 = 0 which is same as variance. Hence proved.

d) p(1, 4) = p(1)p(4) which is true here as there is only one positive value in each row and each column. Therefore m and f are independent variables here.