expectation variance covariance and correlation notes

Chapter Roadmap: Expectation, Variance, and Covariance

Your Journey Through This Chapter 1 Expectation and Linearity The center of mass. LOTUS and the powerful linearity property. 2 Variance and Standard Deviation Measuring the spread. The computational formula and transformation rules. 3 Covariance and Correlation Joint variability. Distinguishing between independence and being uncorrelated. 4 Transformations and Sums Variance of sums, products of independent variables, and linear transformations.

The Heart of Random Variables: Expectation

The Expectation (or expected value, or mean) of a random variable X, denoted as E [X] or μ_{X}, represents the long-run average value or the "center of mass" of its distribution. For Discrete Random Variables: E [X] = x \sum x \cdot P (X = x) For Continuous Random Variables: E [X] = \int_{- \infty}^{\infty} x \cdot f_{X} (x) d x Key Intuition: Expectation is a weighted average. Values with higher probability pull the expectation closer to themselves. It does not necessarily have to be a value that X can actually take (for example, the expected value of a fair die roll is 3.5).

Variance: Measuring the Spread

Variance, denoted as V a r (X) or σ_{X}^{2}, measures the spread or dispersion of the random variable around its mean μ . Definition: V a r (X) = E [(X - μ)^{2}] Computational Formula (Highly Preferred): By expanding the square and using linearity of expectation: V a r (X) = E [X^{2}] - (E [X])^{2} Standard Deviation: σ_{X} = V a r (X) Standard deviation is in the same units as X, making it easier to interpret than variance.

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Expectation, Variance, Covariance and Correlation Notes for GATE DA

Expectation, Variance, Covariance and Correlation notes for GATE DA: 9 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Expectation, Variance, and Covariance

Your Journey Through This Chapter

Expectation and Linearity
The center of mass. LOTUS and the powerful linearity property.

Variance and Standard Deviation
Measuring the spread. The computational formula and transformation rules.

Covariance and Correlation
Joint variability. Distinguishing between independence and being uncorrelated.

Transformations and Sums
Variance of sums, products of independent variables, and linear transformations.

The Heart of Random Variables: Expectation

The Expectation (or expected value, or mean) of a random variable $X$ , denoted as $E [X]$ or $μ_{X}$ , represents the long-run average value or the "center of mass" of its distribution.

For Discrete Random Variables:

E [X] = x \sum x \cdot P (X = x)

For Continuous Random Variables:

E [X] = \int_{- \infty}^{\infty} x \cdot f_{X} (x) d x

Key Intuition: Expectation is a weighted average. Values with higher probability pull the expectation closer to themselves. It does not necessarily have to be a value that $X$ can actually take (for example, the expected value of a fair die roll is $3.5$ ).

Variance: Measuring the Spread

Variance, denoted as $V a r (X)$ or $σ_{X}^{2}$ , measures the spread or dispersion of the random variable around its mean $μ$ .

Definition:

V a r (X) = E [(X - μ)^{2}]

Computational Formula (Highly Preferred):

By expanding the square and using linearity of expectation:

V a r (X) = E [X^{2}] - (E [X])^{2}

Standard Deviation: $σ_{X} = V a r (X)$

Standard deviation is in the same units as $X$ , making it easier to interpret than variance.

Covariance and Correlation Coefficient

Covariance measures the joint variability of two random variables $X$ and $Y$ .

Definition:

C o v (X, Y) = E [(X - μ_{X}) (Y - μ_{Y})]

Computational Formula:

C o v (X, Y) = E [X Y] - E [X] E [Y]

Correlation Coefficient ( $ρ$ ): Covariance is dependent on the units of measurement. To normalize it, we divide by the product of their standard deviations:

ρ_{X, Y} = \frac{C o v ( X , Y )}{σ _{X} σ _{Y}}

$- 1 \leq ρ_{X, Y} \leq 1$
$ρ = 1$ : Perfect positive linear relationship.
$ρ = - 1$ : Perfect negative linear relationship.
$ρ = 0$ : No linear relationship (variables are uncorrelated).