Question

Question 5+ The income and education level of each person on the electoral roll for Queanberra is recorded as a pair (x, y) E 1,2,3)2, where 1 stands for low, 2 for average, and 3 for high, e.g. (2,3) represents a highly educated person with average income. Let S denote the set of all people on the Queanberra electoral roll, and define random variables X, Y : S → { 1,2,3) by X(s),y(s) are the income and educational levels of person 8. let pi,-P(((X(s)-i) Л (Y(s)-J))) for l i, j 3, Assume that 0.02 0.05 0.03 (PJ)isa-0.10 0.25 0.15 0.08 0.20 0.12/ a Explain what pa represents. (b) Find the probability of the event HI: "the person has a high income"? (c) Find the probability of the event LE: "the person has a low education level"? (d) Describe the event HIULE and find its probability. (e) Are the events HI and LE independent? (f) Are the random variables X and Y independent?

Answer 1

a.) p(2,3)

probability of people with average income and high qualification

b.)

probability of high income = sum of probabilities in row 3

= 0.40

c.)

probability of low education = sum of probabilities in column 1

= 0.20

d.)

p(HI U LE) = 0.40+0.20-0.08 as 0.08 is being counted twice

= 0.52

e.)

now if two events are independent there intersection probability must be zero

but here probability of low education with high income = p(3,1)

= 0.08

so these events are not independent

e.)

now for independent X and Y , p(i,j) should be equal to p(j,i)

but its not the case here so X and Y are not independent