Male and female hourly wages have distributions wm N134 and

Male and female hourly wages have distributions wm N(13,4) and wf N(10,9), respectively.

(a) wm and wf denote male and female wages, respectively. If they are independent, what is the distribution of wm + wf ? If the covariance is 6, what is the distribution of 2wm + 3wf ?

(b) What fraction of women has wages above 11? other words, calculate the probability P(wf > 11). To answer this item, you might need to review your lectures on probability and statistics.

Solution

a)

Let Z = 2Wm + 3Wf

Mean Z = 2(13) + 3(10) = 26 + 30 = 56

Var (Z) = 2² Var(Wm) + 3² Var(Wf) = 4(4) + 9(9) = 97

Thus, Z ~ N(56, 97)

If the covariance was 6,

then:

Mean (Z) = 56 (as before)

Var (Z) = 2² Var(Wm) + 3² Var(Wf) + 2 Cov (2Wm, 3Wf) = 4(4) + 9(9) + 2(2) (3) (6)

= 169

Z ~ N(56, 169)

b)

P ( Wf > 11)

= P ( Z > 11 - 10 / sqrt(9) ) where Z = standard normal variate

= P ( Z > 1/3)

= 0.3694

Hope this helps.