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.
[tex]P(X = 0) = _nC_x * P^x * (1 - P)n-x[/tex]
[tex]P(X = 0) = _7C_o * 0.2^0 * (1-0.2)^{7-0[/tex]
[tex]P(X = 0) =1 * 1* (1-0.2)^{7[/tex]
[tex]P(X = 0) =(0.8)^{7[/tex]
[tex]P(X = 0) =0.2097152[/tex]
If it rains on any one day; then X = 100
[tex]P(X = 100) = _nC_x * P^x * (1 - P)n-x[/tex]
[tex]P(X = 100) = _7C_1 * 0.2^1 * (1-0.2)^{7-1[/tex]
[tex]P(X = 0) =7 * 0.2* (1-0.2)^{6[/tex]
[tex]P(X = 100) =7* 0.2* (0.8)^{6[/tex]
[tex]P(X = 100) =0.3670016[/tex]
if it rains on any two day ; then X = 200
[tex]P(X = 200) = _nC_x * P^x * (1 - P)n-x[/tex]
[tex]P(X = 200) = _7C_2 * 0.2^2 * (1-0.2)^{7-2[/tex]
[tex]P(X = 200) = 21 * 0.2^2 * (0.8)^{5[/tex]
[tex]P(X = 200) = 0.2752512[/tex]
if it rains on any three day or more than that ; then X = 300
[tex]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[/tex]
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[tex][Var X] =E [X^2] - (E [X])^2[/tex]
where;
[tex]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[/tex]
[tex](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[/tex]
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
