Coupling and Poisson approximation

Show that $X$ is stochastically larger than $Y$ if and only if $\mathbb {E}\left[u(X)\right] \geq \mathbb {E}\left[u(Y)\right]$ for any nondecreasing function $u$ for which the expectations exist.

Let $X$ and $Y$ be Poisson distributed with respective parameters $\lambda$ and $\mu$. Show that $X$ is stochastically larger than $Y$ if $\lambda \geq \mu$.

Show that the total variation distance between two discrete variables $X, Y$ satisfies

Show for discrete random variables $X, Y$ that $\mathbb {P}\left(X = Y\right) \leq 1-\frac{1}{2} d_{\mathrm{TV}}(X, Y)$, where $d_{\mathrm{TV}}$ denotes total variation distance.

Show that equality is possible in the inequality of Exercise (4.12.4) in the following sense. For any pair $X, Y$ of discrete random variables, there exists a pair $X^{\prime }$, $Y^{\prime }$ having the same marginal distributions as $X, Y$ such that $\mathbb {P}\left(X^{\prime } = Y^{\prime }\right) = 1-\frac{1}{2} d_{\mathrm{TV}}(X, Y)$.

Let $X$ and $Y$ be indicator variables with $\mathbb {E}\left[X\right]= p, \mathbb {E}\left[Y\right] = q$. What is the maximum possible value of $\mathbb {P}\left(X = Y\right)$, as a function of $p, q$? Explain how $X, Y$ need to be distributed in order that $\mathbb {P}\left(X = Y\right)$ be:

The random variable $X$ is said to be bigger than $Y$ in stop-loss order if

for all $a \in \mathbb {R}$, in which case we write $X \geq_{\mathrm{sl}} Y$. Show that $X \geq_{\mathrm{sl}} Y$ is equivalent to: $\mathbb {E}\left[X^{+}\right], \mathbb {E}\left[Y^{+}\right] < \infty$ and $\mathbb {E}\left[c(X)\right] \geq \mathbb {E}\left[c(Y)\right]$ for all increasing convex functions $c$.

The random variable $X$ is said to be smaller than $Y$ in convex order if $\mathbb {E}\left[u(X)\right] \leq \mathbb {E}\left[u(Y)\right]$ for all convex functions $u$ such that the expectations exist, in which case we write $X \leq { }_{\mathrm{cx}} Y$.