Probability
The Analysis of Data, volume 1
Important Random Vectors: Exercises
$
\def\P{\mathsf{\sf P}}
\def\E{\mathsf{\sf E}}
\def\Var{\mathsf{\sf Var}}
\def\Cov{\mathsf{\sf Cov}}
\def\std{\mathsf{\sf std}}
\def\Cor{\mathsf{\sf Cor}}
\def\R{\mathbb{R}}
\def\c{\,|\,}
\def\bb{\boldsymbol}
\def\diag{\mathsf{\sf diag}}
\def\defeq{\stackrel{\tiny\text{def}}{=}}
$
5.7. Exercises
- Justify the derivations of the expectation and variance of the Bernoulli random vector.
- Consider a multinomial vector $\bb X=(X_1,\ldots,X_n)$ and a mapping $\bb X\to Y$ defined by $Y=\sum_{i\in A} X_i$ for some $A\subset \{1,\ldots,n\}$. What is the distribution of $Y$? Write down the pmf in a compact form.
- Characterize the elliptical contours of the multivariate Gaussian pdf with non-diagonal $\Sigma$ in terms of the eigenvalues and eigenvectors of $\Sigma$. Hint: use spectral decomposition (Proposition C.3.8) and the relationship in Proposition 5.2.4.
- Express the exponential, Poisson RVs and the multinomial random vector as exponential family random vectors.