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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.

Respuesta :

Answer:

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

Step-by-step explanation:

Given :

Moment generating function.

[tex]m_D(u)=E(e^{uD})=e^{18u^2+10u}...(1)[/tex]

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

The general moment generating function is :

[tex]m_D(u)=E(e^{uD})=e^{\frac{1}{2}\sigma^2u^2+\mu u}...(2)[/tex]

(a)

Now, compare the equation [tex](1)[/tex] and [tex](2)[/tex] we get,

[tex]\mu=10[/tex]

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

(b)

Now, compare the equation [tex](1)[/tex] and [tex](2)[/tex] we get,

[tex]\frac{1}{2}\sigma^2=18[/tex]

[tex]\Rightarrow \sigma^2=18\times 2=36[/tex]

[tex]\Rightarrow \sigma=\sqrt{36}=6[/tex]

Therefore, the standard deviation of that distance is [tex]6[/tex] miles.

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