Expected Value and Variance

The expected value (E[X]) is the mean of a random variable, representing the long-run average. For a discrete variable, E[X] = Σ x × P(X=x). For a continuous variable, E[X] = ∫ x × f(x) dx. Variance (Var[X]) measures the spread of a distribution: Var[X] = E[(X-μ)²] = E[X²] - μ². The standard deviation is the square root of the variance. Covariance measures the relationship between two variables.

Governing FormulaE[X] = Σ x_i × P(x_i), Var[X] = E[(X-μ)²] = E[X²] - μ², Cov(X,Y) = E[(X-μ_x)(Y-μ_y)]

Knowledge Check

10 Questions

1.A random variable X has values 1, 2, 3 with probabilities 0.2, 0.5, 0.3 respectively. What is E[X]?

2.For the random variable in the previous question, what is Var[X]? (E[X²] = 1×0.2 + 4×0.5 + 9×0.3 = 4.9)

3.What is the variance of a constant value c?

4.If X and Y are independent random variables, what is Var(X+Y)?

5.The covariance between two variables is a measure of:

6.A random variable has E[X²] = 25 and E[X] = 4. What is Var[X]?

7.What is the expected value of the sum of two random variables?

8.The standard deviation is the square root of the:

9.A game pays $10 with probability 0.2, $5 with probability 0.5, and $0 with probability 0.3. What is the expected payout?

10.If Var(X) = 16, what is the standard deviation of X?