Independent solution

How to solve this Multinomial Distribution question

Setup

Setup

Let I and J count the two nonzero outcome categories among the two trials. Their joint distribution is multinomial, so each count has expectation equal to the trial count times its category probability.

E⁡[I]=2(0.20)=0.40\operatorname{E}[I]=2(0.20)=0.40
E⁡[J]=2(0.10)=0.20\operatorname{E}[J]=2(0.10)=0.20

Model

Model

Express the total loss as a linear combination of the category counts.

L=2I+20JL=2I+20J
E⁡[L]=2E⁡[I]+20E⁡[J]\operatorname{E}[L]=2\operatorname{E}[I]+20\operatorname{E}[J]

Compute

Compute

Substitute the two expected counts. No independence between I and J is needed for this expectation.

E⁡[L]=2(0.40)+20(0.20)\operatorname{E}[L]=2(0.40)+20(0.20)
E⁡[L]=0.80+4.00=4.80\operatorname{E}[L]=0.80+4.00=4.80

Answer

Answer

The expected total loss is 4.80.

E⁡[L]=4.80(E)\boxed{\operatorname{E}[L]=4.80\quad\text{(E)}}