Probability ④ Random Variables, Expected Value, and Variance
Expected value: E(X) = Σ xi×P(X=xi)
Variance: V(X) = E(X²) − {E(X)}²
Standard deviation: σ = √V(X)
Example ①
Expected value of the outcome when a die is rolled once: E(X)=(1+2+3+4+5+6)/6 = 7/2
Expected value of the outcome when a die is rolled once: E(X)=(1+2+3+4+5+6)/6 = 7/2
Example ②
When X takes the values 1,2,3 with probability 1/3 each: E(X)=2、E(X²)=14/3
V(X) = 14/3 − 4 = 2/3
When X takes the values 1,2,3 with probability 1/3 each: E(X)=2、E(X²)=14/3
V(X) = 14/3 − 4 = 2/3
Key Points
- E(aX+b) = aE(X)+b
- V(aX+b) = a²V(X)
Practice Problems
- When X takes the values −1,0,1 with probabilities 1/4, 1/2, 1/4, find E(X) and V(X).
- When E(X)=3 and V(X)=4, what are the expected value and variance of Y=2X−1?
- Two coins are tossed. Find the expected number X of heads.
Answers
- E(X)=0、V(X)=1/2
- E(Y)=5、V(Y)=16
- P(0)=1/4,P(1)=1/2,P(2)=1/4 → E(X)=1