A researcher is testing the effectiveness of a political video. She has randomly sampled 120 people to test the video. For each person, there is 50% chance the person will be Democrat and 50% chance the person will be Republican. Republicans have 75% chance of reacting to the video and Democrats have 25% chance of reacting to the video.
What is the expected value and variance of the number of people in the $ 120 $ who respond to the video?
Expected value is straight forward: $ (120*0.5*0.25)+(120*0.5*0.75)=60 $
How do I go about finding the variance though? Is it $ 0 $?
Let $Z$ be the number of people that react to the video. Then we can write $Z=X+Y$, where $X$ is the number of Republican subject that react and $Y$ the number of Democrat subjects. For a fixed number $K=k$ of Republican test subjects, $X$ and $Y$ are independent and have the following distributions: $X|k\sim Bin(k, 0.75), Y|k\sim Bin(120-k, 0.25).$
We then have the conditional expectation $E(Z|K)=0.75K+(120-K)0.25=30+0.5K$. We also know the distribution of the number of Republican subjects: $K\sim Bin(120, 0.5)$. We then have the total expectation: $E(Z)=E(E(Z|K))=E(30+0.5K)=30+0.5E(K)=90.$ The variance we can find using the law of total variance in a similar way.