Answer:
the variance of the refund payment to the couple = 9463.394
Step-by-step explanation:
Given that :
A couple book a cruise to Alaska that promises to refund 100 per day of rain on the seven day cruise up to a maximum of 300.
It is possible that the couple won't be able to refund up 100 per day or more than 100 per day.
SO; let assume that the refund payment happens to be 0, 100,200, 300
Let X be the total refund payment on the seven day cruise.
We can say X = 0, if there is no rain on all 7 days.





If it rains on any one day; then X = 100





if it rains on any two day ; then X = 200




if it rains on any three day or more than that ; then X = 300
![P(X \ge 300) = 1 - P(X < 300) \\ \\ P(X \ge 300) = 1 - [P(X = 0) + P(X = 100) + P(X = 200)] \\ \\ P(X \ge 300) = 1 - [0.2097152 + 0.3670016 + 0.2752512] \\ \\ P(X \ge 300) = 0.148032](https://tex.z-dn.net/?f=P%28X%20%5Cge%20300%29%20%3D%201%20-%20P%28X%20%3C%20300%29%20%20%5C%5C%20%5C%5C%20P%28X%20%5Cge%20300%29%20%3D%201%20-%20%5BP%28X%20%3D%200%29%20%2B%20P%28X%20%3D%20100%29%20%2B%20P%28X%20%3D%20200%29%5D%20%5C%5C%20%5C%5C%20P%28X%20%5Cge%20300%29%20%3D%201%20-%20%5B0.2097152%20%2B%200.3670016%20%2B%200.2752512%5D%20%5C%5C%20%5C%5C%20P%28X%20%5Cge%20300%29%20%3D%200.148032)
Now; we have our probability distribution function as:
P(X = 0) = 0.2097152
P(X = 100) = 0.3670016
P(X = 200) = 0.2752512
P(X = 300) = 0.148032
In order to determine the variance of the refund payment to the couple; we use the formula:
variance of the refund payment to the couple![[Var X] =E [X^2] - (E [X])^2](https://tex.z-dn.net/?f=%5BVar%20X%5D%20%3DE%20%5BX%5E2%5D%20-%20%28E%20%5BX%5D%29%5E2)
where;
![E[X^2] = \sum x^2 \times p \\ \\ E[X^2] = 0^2 * 0.2097152 + 100^2 * 0.3670016 + 200^2 * 0.2752512 + 300^2 * 0.148032 \\ \\ E[X^2] = 0 + 3670.016 + 11010.048+ 13322.88 \\ \\ E[X^2] =28002.944](https://tex.z-dn.net/?f=E%5BX%5E2%5D%20%20%3D%20%5Csum%20x%5E2%20%5Ctimes%20p%20%5C%5C%20%5C%5C%20E%5BX%5E2%5D%20%20%3D%200%5E2%20%2A%200.2097152%20%2B%20100%5E2%20%2A%200.3670016%20%2B%20200%5E2%20%2A%200.2752512%20%2B%20300%5E2%20%2A%200.148032%20%5C%5C%20%5C%5C%20%20E%5BX%5E2%5D%20%20%3D%200%20%20%2B%203670.016%20%2B%2011010.048%2B%2013322.88%20%20%5C%5C%20%5C%5C%20%20E%5BX%5E2%5D%20%20%3D28002.944)
![(E [X]) = \sum x * p\\ \\ (E [X]) = 0 * 0.2097152 + 100 * 0.3670016 + 200 * 0.2752512 + 300 * 0.148032 \\ \\ (E [X]) = 0 + 36.70016 + 55.05024 + 44.4096\\ \\ (E [X]) = 136.16 \\ \\ (E [X])^2 = 136.16^2 \\ \\ (E [X])^2 = 18539.55](https://tex.z-dn.net/?f=%28E%20%5BX%5D%29%20%3D%20%5Csum%20x%20%2A%20p%5C%5C%20%5C%5C%20%20%28E%20%5BX%5D%29%20%3D%20%200%20%2A%200.2097152%20%2B%20100%20%2A%200.3670016%20%2B%20200%20%2A%200.2752512%20%2B%20300%20%2A%200.148032%20%5C%5C%20%5C%5C%20%28E%20%5BX%5D%29%20%3D%200%20%2B%2036.70016%20%2B%2055.05024%20%2B%2044.4096%5C%5C%20%5C%5C%20%28E%20%5BX%5D%29%20%3D%20136.16%20%5C%5C%20%5C%5C%20%28E%20%5BX%5D%29%5E2%20%3D%20136.16%5E2%20%5C%5C%20%5C%5C%20%28E%20%5BX%5D%29%5E2%20%3D%2018539.55)
NOW;
the variance of the refund payment to the couple = 28002.944 - 18539.55
the variance of the refund payment to the couple = 9463.394