Question

The ancient Temple of Thos is believed to be buried near the north-south directed main canal through the modern city of Gyre. There is considerable uncertainty, though as to where. A probabilist believes that the distance D of the temple in miles north from the center of Gyre follows a moment generating function

mD(mu) = E(euD) = e18u2 + 10 u
a) Find the expected value of the distance north from city center archaeologists will have to travel to find the temple.
b) Find the standard deviation of that distance.

Answer 1

Hence, the expected value of the distance north from city centre archaeologists will have to travel to find the temple is
`10` miles and the standard deviation of that distance is
`6` miles.

Explanation:

Given :

Moment generating function.

`m_D(u)=E(e^(uD))=e^(18u^2+10u)...(1)`

Any two functions can not have the same moment generating function.

The general moment generating function is :

`m_D(u)=E(e^(uD))=e^{(1)/(2)\sigma^2u^2+\mu u}...(2)`

(a)

Now, compare the equation
`(1)` and
`(2)` we get,

`\mu=10`

Therefore, the expected value of the distance north from city centre archaeologists will have to travel to find the temple is
`10 miles.`

(b)

Now, compare the equation
`(1)` and
`(2)` we get,

`(1)/(2)\sigma^2=18`

`\Rightarrow \sigma^2=18* 2=36`

`\Rightarrow \sigma=√(36)=6`

Therefore, the standard deviation of that distance is
`6` miles.