Final answer:
The standard deviation of the amount the insurance company will have to pay can be calculated using the Poisson distribution, with the mean of consecutive days of rain being 0.6. The payment amounts are capped, which influences the calculation of the standard deviation of the payment amount.
Step-by-step explanation:
The question involves calculating the standard deviation of the amount the insurance company will have to pay based on a Poisson random variable for the number of consecutive days of rain. Given the mean (λ) of the Poisson distribution is 0.6, the variance of a Poisson distribution is equal to its mean. Therefore, the variance is also 0.6. Since standard deviation is the square root of the variance, the standard deviation of the number of consecutive days of rain is the square root of 0.6.
For the insurance policy, the insurance company will pay $1000 for each day the game is postponed, up to 2 days. We need to consider this cap when we're calculating the standard deviation of the payment amount. There are three possible scenarios: no rain (0 days postponed), rain for one day (1 day postponed), or rain for two or more days (2 days postponed). The payments would thus be $0, $1000, or $2000, respectively. If we denote the random variable for the payment amount by Y, then the standard deviation of Y can be found by calculating the expected value of Y squared (E[Y^2]) minus the square of the expected value of Y (E[Y]) squared, and then taking the square root of the result.
Using the probabilities provided by the Poisson distribution for 0, 1, and 2 or more days, we can calculate the expected value and the variance of the payment amount. Then, the standard deviation would be the square root of the calculated variance of the payment amount.