A field of geysers in Yellowstone may display an eruption between once every 45 minutes, to as much as every 13 hours. Assume the probability is a uniform continuous probability distribution. Find the theoretical average time between eruptions for geysers in this field, and the theoretical standard deviation for these times.

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Answer to a math question A field of geysers in Yellowstone may display an eruption between once every 45 minutes, to as much as every 13 hours. Assume the probability is a uniform continuous probability distribution. Find the theoretical average time between eruptions for geysers in this field, and the theoretical standard deviation for these times.

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Cristian

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119 Answers

1. The eruption times are uniformly distributed between 45 minutes and 13 hours.
2. Convert 45 minutes to hours:

45 \text{ minutes} = \frac{45}{60} = 0.75 \text{ hours}

.
3. The random variable

representing eruption times is uniformly distributed on the interval

[0.75, 13]

.
4. The average (mean) of a uniform distribution

is calculated as

\frac{a + b}{2}

\text{Mean} = \frac{0.75 + 13}{2} = \frac{13.75}{2} = 6.875

hours.

5. The standard deviation of a uniform distribution from

is calculated as

\frac{b-a}{\sqrt{12}}

\text{Standard Deviation} = \frac{13 - 0.75}{\sqrt{12}} = \frac{12.25}{\sqrt{12}} = \frac{12.25}{3.464} \approx 3.5367

hours.

Therefore, the mean eruption time is

6.875 \text{ hours}

, and the standard deviation is

3.5367 \text{ hours}

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