Probability

The Analysis of Data, volume 1

Important Random Vectors: Exercises

5.7. Exercises

1. Justify the derivations of the expectation and variance of the Bernoulli random vector.
2. 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.
3. 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.
4. Express the exponential, Poisson RVs and the multinomial random vector as exponential family random vectors.