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

Answer:

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.

Step-by-step 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 :

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

(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,

$\frac{1}{2}\sigma^2=18$

$\Rightarrow \sigma^2=18\times 2=36$

$\Rightarrow \sigma=\sqrt{36}=6$

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