Theorem 3.5.1 (Chebychev's Inequality) Let $X$ be a random variable and let $g(x)$ be a nonnegative function. Then, for any $r>0,$ \[P(g(X)\geq r) \leq \frac{\E[g(X)]}{r}.\]

Theorem 3.5.2 (Markov's Inequality) If $P(X \geq 0) = 1$ and $P(X = 0) < 1$, then, for any $r>0,$ \[P(X \geq r) \leq \frac{\E(X)}{r}\] with equality if and only if $P(X = r) = p = 1-P(X = 0),\; 0<p\leq 1.$

Theorem 3.5.3 Let $X_{\alpha,\beta}$ denote a $\text{gamma}(\alpha,\beta)$ random variable with pdf $f(x \vert \alpha,\beta),$ where $\alpha > 1$. Then for any constants $a$ and $b$, \[P(a < X_{\alpha,\beta} < b) = \beta(f(a \vert \alpha, \beta) - f(b\vert \alpha, \beta)) + P(a<X_{\alpha-1,\beta} < b).\]

Lemma 3.5.4 (Stein's Lemma) Let $X \sim n(\theta,\sigma^2),$ and let $g$ be a differentiable function statisfying $\E\vert g^{\prime}(X)\vert < \infty.$ Then \[\E[g(X)(X-\theta)]= \sigma^2\E[g^{\prime}(X)].\]

Theorem 3.5.5 Let $\mathcal{X}_p^2$ deonte a chi-squared random variable with $p$ degrees of freedom. For any function $h(x),$ \[\E[h(\mathcal{X}_p^2)] = p\E\left(\frac{h(\mathcal{X}_{p+2}^2)}{\mathcal{X}_{p+2}^2}\right)\] provided the expectations exist.

Theorem 3.5.6 (Hwang) Let $g(x)$ be a function with $-\infty < \E[g(X)] < \infty$ and $-\infty < g(-1) < \infty.$ Then:
a. If $X \sim \text{Poisson}(\lambda),$ \[\E[\lambda g(X)] = \E[Xg(X-1)].\] b. If $X \sim \text{NegBinom}(r,p),$ \[\E[(1-p)g(X)] = \E\left(\frac{X}{r+X - 1}g(X-1)\right).\]