Question

Consider a joint PMF for the results of a study that compared the number of micro-strokes a patient suffered in a year (F) and an index (S) that characterizes the stress the person is exposed to. This PMF represents the probability of a randomly picked person from the studied population having F=f micro-strokes and S=s stress index.

	f=0	f=1	f=2	f=3
s=1	0.1	0.04	0.04	0.02
s=2	0.25	0.1	0.12	0.03
s=3	0.15	0.06	0.03	0.06

a) The conditional PMF for the number of strokes F given stress index S=3.


b) The expected number of strokes and the variance of this magnitude for patients with S=3?


c) The conditional PMF for strokes and stress index given event A={(S,F) /s<3 and f<2}


d) There were 3000 patients in the study. How many you expect to find that have F and S in A (same A as above)?


e) What is the average stress index in this population? (hint: the marginal probability function above may be helpful)

Answer 1

	f=0	f=1	f=2	f=3	P(s)
s=1	0.1	0.04	0.04	0.02	0.2
s=2	0.25	0.1	0.12	0.03	0.5
s=3	0.15	0.06	0.03	0.06	0.3
P(f)	0.5	0.2	0.19	0.11	1

a.)

Conditional PMF of f given s = 3 is given by

P(f | s =3) = P (f ,3) / P(s = 3)

P(s = 3 ) = 0.15 + 0.06 + 0.03 + 0.06 = 0.3

P ( f = 0 | s = 3) = 0.15/0.3 = 0.5

P ( f = 1 | s = 3) = 0.06/0.3 = 0.2

P ( f = 2 | s = 3) = 0.03/0.3 = 0.1

P ( f = 3 | s = 3) = 0.06/0.3 = 0.2

	f=0	f=1	f=2	f=3
P(f|s=3)	0.5	0.2	0.1	0.2	1

b.)

Expected number of strokes for patients with s = 3

E( f | s = 3 ) =

= 0*0.5 +1*0.2 + 2*0.1 + 3*0.2 = 1

E( f | s = 3 ) = 1.

Hence, Expected number of strokes for patients with s = 3 is 1 stroke.

= (0²*0.5 + 1²*0.2 + 2²*0.1 + 3²*0.2 ) - 1 = 0 + 0.2 + 0.4 + 1.8 - 1 = 2.4 - 1 = 1.4

c.)

A={(S,F) /s<3 and f<2}

for s<3 this implies s= 1,2

for f<2 this implies f= 0,1

P(f = 0) = 0.1 + 0.25 + 0.15 = 0.5

P(f = 1) = 0.04 + 0.1 + 0.06 = 0.2

P(s = 2) = 0.25 + 0.1 + 0.12 + 0.03 = 0.19

P (s = 1) = 0.1 + 0.04 + 0.04 + 0.02 = 0.11

F|S

P(f =0 | s =1) = P (f=0,s=1) / P(s = 1) = 0.1/0.2 = 0.5.

P(f =1 | s =1) = P (f=1,s=1) / P(s = 1) = 0.04/0.2 = 0.2.

P(f =0 | s =2) = P (f=0,s=2) / P(s = 2) = 0.25/0.5 = 0.5.

P(f =1 | s =2) = P (f=1,s=2) / P(s = 2) = 0.1/0.5 = 0.2.

Conditional PMF of

P(f<2|s<3)

	F|S
P(f<2|s<3)	f=0	f=1
s=1	0.5	0.2
s=2	0.5	0.2

S|F

P( s= 1 | f = 0) = P (f=0,s=1) / P(f = 0) = 0.1/0.5 = 0.2

P( s= 2 | f = 0) = P (f=0,s=2) / P(f = 0) = 0.25/0.5 = 0.5

P( s= 1 | f = 1) = P (f=1,s=1) / P(f = 1) = 0.04/0.2 = 0.2

P( s= 2 | f = 1) = P (f=1,s=2) / P(f = 1) = 0.1/0.2 = 0.5

	S|F
P(s<3|f<2)	f=0	f=1
s=1	0.2	0.2
s=2	0.5	0.5

d.)

No of patients expected = total number of patients in the study * required probability

Required probability = P(s<3,f<2) = 0.1 + 0.04 + 0.25 + 0.1 = 0.49

If there were 3000 patients in the study we expect to have 1470 patients in F and S in A (same A as above).

e.)

s	P(s)	s*p(s)
1	0.2	0.2
2	0.5	1
3	0.3	0.9
	1	2.1

The probabilities calculated in the last column of the table above are marginal probabilities.

Average stress index = E(S) =   = 1*0.2 + 2*0.5 + 3*0.3 = 2.1

Therefore, Average index in this population is 2.1.