a) The table below shows the probability distribution, where P is probability:
RAINFALL |
P |
Profit (RM) |
Heavy |
0.3 |
28000 |
Moderate |
0.35 |
55000 |
Little |
0.2 |
15000 |
None |
0.15 |
7000 |
Consider how the meteorologist got his figures. Let's say he measured rainfall over a period of 365 days and discovered the following:
No rainfall for 54.75 days
Little rainfall for 73 days
Moderate rainfall for 127.75 days
Heavy rainfall for 109.5 days
These periods of time would give the probabilities quoted.
Based on these, and using a prorata value for the farmer's profit, we can calculate his profit for these periods:
54.75/365*7000+73/365*15000+127.75/365*55000+109.5/365*28000=31700. So the answer to b) (your second a) is RM31700.
No rainfall for the whole year produces a profit of RM7000; but little or no rainfall requires adjustment. The time period is 54.75+73=127.75 which is 0.35 year. So the profit has to be extrapolated based on the factor 1/0.35. So we have (54.75/365*7000+73/365*15000)/0.35=RM11571.43. For rainfall ranging from none to moderate, the time period is 0.7 year. The adjustment or extrapolation factor is 1/0.7. This gives us 23300/0.7=RM33285.71. And, of course, all types of rainfall give us RM31700. The tables below show cumulative probability.
Working from heavy to no rainfall:
RAINFALL |
P |
Profit (RM) |
Heavy |
0.3 |
28000 |
Heavy/moderate |
0.65 |
42538.46 |
Heavy/moderate/little |
0.85 |
36058.82 |
All |
1 |
31700 |
And from no to heavy rainfall:
RAINFALL |
P |
Profit (RM) |
None |
0.15 |
7000 |
None/little |
0.35 |
11571.43 |
None/little/moderate |
0.7 |
33285.71 |
All |
1 |
31700 |
In both the last two tables we have datasets for which we can calculate mean and SD for 7 rainfall patterns:
c) and d) No rainfall at all; little or no rain; all but heavy rain; a mixture of all types; heavy to moderate; some daily rain; only heavy rain. Using figures from the tables we arrive at a mean of RM27164.92 and a SD of RM12094.61, which gives a range for the profit of RM15070.31 to RM39259.53. (SD is square root of VARIANCE. Both mean and variance are calculated by dividing the sums of the relevant columns by 7, the number of rainfall types.)
RAINFALL |
Profit |
Profit-mean |
(Profit-mean)^2 |
None |
7000 |
-20164.92 |
406624000 |
Little or none |
11571.43 |
-15593.49 |
243156930 |
No heavy rain |
33285.71 |
6120.79 |
37464070 |
Mixed |
31700 |
4535.08 |
20566951 |
Heavy or moderate |
42538.46 |
15373.54 |
236345732 |
Some daily rain |
36058.82 |
8893.9 |
79101457 |
Only heavy rain |
28000 |
835.08 |
697359 |
MEAN: |
27164.92 |
VARIANCE: |
146279500 |