Bob earns $80 per day, Monday to Friday inclusive. He works every alternate Saturday for which he earns 'time and a half's and every fourth Sunday, for which he is paid 'double time'.
Show that there are two possible patterns Bob could work over a typical four-week period, depending on which Saturdays and Sunday he works. Hence find the expectation and variance of his weekly wage under either pattern.
Let's denote Bob's earnings for a typical 4-week period as W.
Pattern 1:
- Monday to Friday: $80 * 5 days = $400
- Every alternate Saturday: $80 * 1.5 = $120
- Every fourth Sunday: $80 * 2 = $160
- Total earnings for 4 weeks = $400 * 5 + $120 * 2 + $160 = $2200
Pattern 2:
- Monday to Friday: $80 * 4 days = $320
- Every alternate Saturday: $80 * 1.5 = $120
- Every fourth Sunday: $80 * 2 = $160
- Total earnings for 4 weeks = $320 * 5 + $120 * 3 + $160 = $2200
Therefore, there are two possible patterns of Bob's work over a typical 4-week period, and his expected weekly wage under either pattern is $550 (since $2200/4 weeks = $550/week).
To find the variance of his weekly wage, we need to calculate the variance of his earnings per day first.
Variance of earnings per day:
- Using Monday to Friday earnings only:
We need to calculate the variance of the amount Bob earns per day during the week (Monday to Friday). The earnings for these days are all the same ($80), so the variance of earnings per day is 0.
- Using all days (including Saturdays and Sundays):
Var(X) = E(X^2) - [E(X)]^2 = (80^2 * 5 + 120^2 * 2 + 160^2) / 7 - (2200/7)^2
Variance of Bob's weekly wage = 7 * Var(X)
By substituting the values above, we can calculate the variance of Bob's weekly wage.