If X takes values 0 and 1 with probabilities 0.2 and 0.8, respectively, what is E[X]?

• A. 0.2
• B. 0.5
• C. 0.8
• D. 1.0

Answer: (C)

The main idea is that the expected value is the probability-weighted average of the possible values. Here X can be 0 with probability 0.2 and 1 with probability 0.8. Multiply and add: E[X] = 0×0.2 + 1×0.8 = 0.8. So the long-run average value of X is 0.8. Since this is a Bernoulli-type variable, E[X] equals the probability of getting 1, which is 0.8. The other numbers would come from different probability placements: 0.2 would occur if the chance of 1 were 0.2, 0.5 would be if the two outcomes were equally likely, and 1.0 would mean X is always 1.