Hassan Ijaz
Ai, Web & Design
Expectation, variance, and covariance
Visual scatter plot generator where users can drag points and see mean, variance, and covariance update in real-time with geometric interpretations
Concept Overview
Expectation (mean) and variance are fundamental measures that characterize probability distributions. The expectation tells us the "center" of a distribution, while variance measures its "spread". Covariance extends these concepts to measure relationships between variables.
Expectation (Expected Value)
The expectation is the theoretical average value of a random variable over many repetitions.
Discrete: E[X] = Σ x · P(X = x)
Sum of each value times its probability
Continuous: E[X] = ∫ x · f(x) dx
Integral of value times density
Key Properties: E[aX + b] = aE[X] + b, E[X + Y] = E[X] + E[Y]
Variance
Variance measures the average squared deviation from the mean - how spread out the values are.
Var(X) = E[(X - μ)²] = E[X²] - (E[X])²
Average squared distance from mean
SD(X) = σ = √Var(X)
Standard deviation - in original units
Key Properties: Var(aX + b) = a²Var(X), Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y)
Covariance
Covariance measures how two variables change together - positive means they tend to increase together, negative means one increases as the other decreases.
Cov(X,Y) = E[(X - μ_X)(Y - μ_Y)] = E[XY] - E[X]E[Y]
Average product of deviations from means
Corr(X,Y) = ρ = Cov(X,Y) / (σ_X · σ_Y)
Correlation: normalized to [-1, 1]
Geometric Interpretation
Mean as Balance Point
The mean is where the distribution would balance if values were weights
Variance as Spread
Larger variance means data is more spread out from the center
Covariance as Tilt
Shows direction and strength of linear relationship
Important Special Cases
- Independent Variables:If X and Y are independent, then Cov(X,Y) = 0 and Var(X+Y) = Var(X) + Var(Y)
- Linear Combinations:E[aX + bY] = aE[X] + bE[Y] always holds
- Perfect Correlation:ρ = ±1 means perfect linear relationship
Practical Example: Portfolio Risk
For a portfolio with two assets:
- Expected return: weighted average of individual returns
- Risk (variance): depends on individual variances AND covariance
- Negative correlation between assets reduces overall risk!
The interactive visualization below lets you drag points to see how mean, variance, and covariance update in real-time. Notice the geometric interpretations as you move the data points around.
Interactive Visualization
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Visual scatter plot generator where users can drag points and see mean, variance, and covariance update in real-time with geometric interpretations