Some Useful Inequalities
This is a list of useful mathematical inequalities. Mostly this is for my own benefit since I can never seem to remember any of them. The list is very small right now but I'll try to add more as I use them. The syntax is LaTeX but hopefully your browser is properly displaying the MathML generated by LaTeXMathML.
Probability
- Markov's Inequality: For non-negative random variable $X$, $Pr\{X \geq a\} \leq \frac{\mu_X}{a}$
- Chebyshev's Inequality: For non-negative $k$, $Pr\left\{|X-\mu_X| \geq k\sigma_X\left\} \leq \frac{1}{k^2}$
Summations
- $\ln{(n+1)} \leq \sum_{i=1}^n \frac{1}{i} \leq \ln{(n)} + 1$
- Jensen's Inequality: For a convex function $f$, $f\left(\sum_{i=0}^n \frac{x_i}{n}\right) \leq \sum_{i=0}^n \frac{f(x_i)}{n}$
Miscellaneous
- $\left(1-\frac{1}{n}\right)^n < \frac{1}{e}$
- $1+x \leq e^x$
- Stirling's Formula: $n! = \sqrt{2\pi n} \left(\frac{n}{e}\right)^n e^{\lambda_n}$, where $\frac{1}{12n+1} < \lambda_n < \frac{1}{12n}$
- Stirling's Formula Simplified: $2\sqrt{n}\left(\frac{n}{e}\right)^n < n! < 3\sqrt{n}\left(\frac{n}{e}\right)^n$
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