Question

Given the following probability distributions for variables X and Y:

P(x, y)X                  Y

0.4       100            200

0.6       200            100

a. E(X) and E(Y).

b. σX and σY.

c. σXY. d. E(X + Y).

e. Suppose that X represents the number of patients successfully treated for Malaria and Y represents the number of patients successfully treated for Tuberculosis. And medication A (first row in the table) has a 40% of effectiveness and medication B (second row in the table) has a 60% of effectiveness. Interpret and make statements based on the calculations you did.

Answer 1

First we construct the joint probability table and find the marginal probabilities of X and Y

		Y	Y
		100	200	P(X)
X	100	0	0.4	0.4
X	200	0.6	0	0.6
	P(Y)	0.6	0.4	1

a) E(X) =

= 100*0.4 +200*0.6

= 160

E(Y) =

= 100*0.6 +200*0.4

= 140

b)

=

= 48.99

=

= 48.99

c)

  

= 100^2*200^2*0.4 + 200^2*100^2*0.6

= 40000000

= 141.42

d) E(X+Y) = E(X) +E(Y) = 140+160 = 300

e) E(X) =160 means average  number of patients treated for Maleria using medication  A or B is 160

E(Y) =140 means average  number of patients treated for Tuberculosis  using medication  A or B is 140

means average distance between the number of patients treated and mean is 48.99

means average distance between the number of patients treated and mean is 48.99

means average distance of joint distribution of number of patients from their means is 141.42

E(X+Y) is the average number of patients treated for Maleria and Tuberculosis is using A or B is 300