Question

Question: Use the law of total expectations to find the average weight of breeding elephant seal, given that 12% of the breeding elephant seals are male and the rest are female, and the expected weight of the breeding elephant seal is 4,200 pounds for a male and 1,100 pounds for a female.

Short answer

Answer

The Average weight of Breeding elephant seal is 1472 pounds.

Step-by-step solution

Given Information

12% of the breeding elephant seals are male and the rest are female, and the expected weight of the breeding elephant seal is 4,200 pounds for a male and 1,100 pounds for a female.

Concept used

The law of total expectation, also known as the law of iterated expectations (or LIE) and the “tower rule”.

\(\sum\limits_{i = I}^n {E\left( {X|{S_i}} \right)} \cdot P\left( {{S_i}} \right)\)

Calculating the average weight of breeding Elephant

Let\({S_1}\)be the male and \({S_2}\)be the female.

We have,

\(\begin{array}{l}P\left( {{S_1}} \right) = 12\% = 0.12\\P\left( {{S_2}} \right) = 1 - P\left( {{S_1}} \right)\\ = 1 - 0.12\\ = 0.88\end{array}\)

Let X be the weight of breeding elephant seal, then

\(\begin{array}{l}E\left( {X|{S_1}} \right) = 4200\\E\left( {X|{S_2}} \right) = 1100\end{array}\)

Using the law of Expectation, we get

\(\begin{array}{l}E\left( X \right) = E\left( {X|{S_1}} \right) \cdot P\left( {{S_1}} \right) + E\left( {X|{S_2}} \right) \cdot P\left( {{S_2}} \right)\\ = 4200\left( {0.12} \right) + 1100\left( {0.88} \right)\\ = 1472\end{array}\)

Thus, the Average weight of Breeding elephant seal is 1472 pounds.