Lecture 2

Mean and variance: μx = E(X) and σx2 = Var(X) = E(X − μx)2

Skewness (symmetry) and kurtosis (fat-tails)

• Kurtosis: How high is the p
• High kurtosis implies heavy (or long) tails in dis- tribution.
• Symmetry has important implications in holding short or long financial positions and in risk man- agement.

(X − μx)3 (X − μx)4 S ( x ) = E σ x3 , K ( x ) = E σ x4 .

Normal distribution

E(X) = μ Var(X) = σ2 S(X) = 0 K(X) = 3 ml = 0, for l is odd.

T-distribution

Symmetry at 0

\[E(x) > 0, v >1\]

Chi-squared distribution

\[E(X) = k$$ $$Var(X) = 2k\]

Joint Distribution

\[F_{X,Y}(x, y) = P(X\leq x, Y\leq y)\]

Marginal Distribution

The marginal distribution of X is obtained by integrating out Y . A similar definition applies to the marginal distribution of Y .