# Finding variance without the data

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 $$0$$?

• You can get the variance from the probability value. However, you need to first define "variance of what?" and "expected value of what?". – NoChance Feb 28 at 7:27
• Each sampled person has a $50\%$ probability of reacting to the video so this a simple binomial question – Henry Feb 28 at 7:32
• @NoChance Let Y be the number of people in the 120 who respond to political video. What is the expected value of Y and variance of Y? – Peter Andrews Feb 28 at 7:36

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.