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.

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 X representing eruption times is uniformly distributed on the interval [0.75, 13].
4. The average (mean) of a uniform distribution a to b 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 a to b 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}.