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

The probabilities are zero for all other values of S and F.

a) Show the table above represents a proper PMF.

b) Calculate the marginal probabilities for S and F and plot.

c) What is the probability a person in the study has had 0 micro-strokes and what is the probability a person in the study has had at least one micro-stroke?

d) What is the probability a person in the study has stress index 2 or above and 2 or more micro-strokes? Are S and F independent? Justify

Answer 1

a)

	f=0	f=1	f=2	f=3
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
	0.5	0.2	0.19	0.11	1

since each probability >= 0

ans sum of probability = 1

hence this is valid pmf

b)

s	p
1	0.2
2	0.5
3	0.3
f	p
0	0.5
1	0.2
2	0.19
3	0.11

c)

P(f = 0) = 0.5

P(f >= 1) = 1- P(f = 0) = 0.5

d)

P(S >= 2 and f >= 2)

= 0.12 + 0.03 + 0.03 + 0.06

= 0.24

P(S = 1)= 0.2

P(f = 2) = 0.19

P(s = 1 , f = 2) = 0.04

since

P(s = 1 , f = 2) P(S = 1) * P(f = 2)

S and F are not independent

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