Motivation: Same Mean, Different Risk
Consider three scenarios where you win money, all with expected value of $1000:
Interactive: Spread Visualizer
Scenario A: You get $1000 guaranteed.
Expected Value
$1000
Variance
0
Std. Deviation
$0
Mean as a Balance Point
The expected value (mean) acts as the "balance point" of a distribution. Variance measures how far values deviate from this point.
Interactive: Balance Point Visualizer
Mean: 3.5
Definition
Variance
$$ \text{Var}(X) = E[(X - \mu)^2] = \sum_x (x - \mu)^2 p(x) $$ where \(\mu = E[X]\).
Standard Deviation
$$ \sigma = \sqrt{\text{Var}(X)} $$
Interactive Proof: Computational Formula
We will prove that: \(\text{Var}(X) = E[X^2] - (E[X])^2\)
Step 1: Start with the definition of variance:
$$ \text{Var}(X) = E[(X - \mu)^2] $$
Step 2: Expand the square:
$$ \text{Var}(X) = E[X^2 - 2\mu X + \mu^2] $$
Step 3: Use linearity of expectation:
$$ \text{Var}(X) = E[X^2] - 2\mu E[X] + \mu^2 $$
Step 4: Since \(\mu = E[X]\):
$$ \text{Var}(X) = E[X^2] - 2\mu^2 + \mu^2 $$
Step 5: Simplify:
$$ \text{Var}(X) = E[X^2] - \mu^2 = E[X^2] - (E[X])^2 \quad \blacksquare $$
Variance Calculator
Enter a custom probability distribution and see its variance computed step-by-step.