QFC – Chapter 3: Probability Theory (10-Liner)

1. Random variables and probability distributions form the foundation of statistical inference in financial markets.

2. The \(\chi_n^2\) distribution has expectation \(n\) and variance \(2n\).

3. Zero covariance does not imply independence, as illustrated by \(X\) and \(X^2\).

4. Probability density functions (PDFs) and cumulative distribution functions (CDFs) characterize continuous distributions.

5. Zero skewness does not necessarily imply symmetry of a distribution.

6. Correlation is invariant under positive affine transformations, but generally not under nonlinear transformations.

7. EUR/USD and GBP/USD exchange-rate returns illustrate positive dependence in currency markets.

8. The Central Limit Theorem motivates Gaussian approximations for sample averages.

9. The multivariate Delta Method derives asymptotic distributions of nonlinear transformations, such as fuel-efficiency conversions.

10. The laws of total expectation and variance establish \({\sf E}[Y]={\sf E}[{\sf E}(Y\mid X)]\) and \(\operatorname{Var}(Y)={\sf E}[\operatorname{Var}(Y\mid X)]+\operatorname{Var}[{\sf E}(Y\mid X)]\).