Final answer:
To prove the variance equation Var(X) = E(X²) - [E(X)]², expand the square of the definition of variance, apply the properties of expected value, and simplify to show that the variance is the difference between the expected value of the square of X and the square of the expected value of X.
Step-by-step explanation:
To prove Var(x) equals E(x²) minus [E(x)]², we start with the definition of variance. The variance of a random variable X is defined as the expected value of the squared deviations from the mean, which can be written as Var(X) = E[(X - μ)²], where μ is the mean of X. By expanding the square, we get E[(X - μ)²] = E[X² - 2Xμ + μ²]. Using the linearity property of expected value, we can split this into E[X²] - 2μE[X] + μ².
Since μ is a constant, E[X] is just μ, simplifying the expression to E[X²] - 2μ² + μ². This further simplifies to E[X²] - μ², which proves that Var(X) = E[X²] - [E(X)]² because μ is equal to E(X). Hence, the variance of a random variable X is indeed the expected value of X squared minus the square of the expected value of X.