Question

An insurance policy for a Dyson vacuum pays $400 if it fails during the first year, with the payout decreasing by $100 each following year until it reaches zero. If the vacuum is still working at the beginning of any given year, and the probability of it failing during that year is 0.3, calculate the standard deviation of the payout for this Dyson vacuum under this policy.

Answer 1

The standard deviation of the payout for this Dyson vacuum is approximately $13.12.

Step-by-step explanation:

To calculate the standard deviation of the payout for this Dyson vacuum under this policy, we need to first determine the payout for each year. The payout starts at $400 and decreases by $100 each following year until it reaches zero.

Let's calculate the payout for each year:

1. Year 1: $400
2. Year 2: $300
3. Year 3: $200
4. Year 4: $100
5. Year 5: $0

Next, we need to calculate the variance of the payout. The probability of the vacuum failing during a given year is 0.3, so the probability of it not failing is 0.7. We multiply the payout for each year by the probability of that year and square the result. The sums of these squared values give us the variance.

1. Year 1: ($400 * 0.3)² = $36
2. Year 2: ($300 * 0.7)² = $63
3. Year 3: ($200 * 0.7)² = $49
4. Year 4: ($100 * 0.7)² = $24.5
5. Year 5: ($0 * 0.7)² = $0

The variance is the sum of these values, which is $172.5. Finally, we take the square root of the variance to get the standard deviation, which is approximately $13.12.