Inequality
Inequalities
$$\frac{n!}{m!} \leq n^{n-m}$$
Markov inequality
Data processing inequality
Fano's inequality
$$H(X|\hat{X}(Y)) \leq H(X|Y) \leq \Pr(X\neq\hat{X})log(|{}\mathcal{X}|-1) + h(\Pr(X\neq\hat{X})) \leq \Pr(X\neq\hat{X})log|{}\mathcal{X}| + 1$$
log-sum inequality
Triangle inequality
In a normed vector space? V, the triangle inequality is
$$\displaystyle \|x + y\| \leq \|x\| + \|y\| \quad \forall \, x, y \in V$$
Cauchy-schwartz inequality
The Cauchy–Schwarz inequality states that for all vectors x and y of a real number|real? or complex number|complex? inner product space?,
$$|\langle x,y\rangle| \leq \|x\| \cdot \|y\|.$$
(Think of <math> \langle x,y \rangle = \|x\|\|y\|cos\theta </math>)
In euclidean space, :<math>\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).</math>
For the inner product space of square-integrable complex-valued functions
A generalization of this is the Hölder inequality?.
Expectation (probability) is inner product of this form.
Jensen's inequality
If f is a convex function $$ f\left(\mathbb{E}\{X\}\right) \leq \mathbb{E}\{f(X)\}.$$
Chebyshev inequality
mean から離れた値の確率は低い。どの確率分布でも。 No more than <math>1/k^2</math> of the values are more than k standard deviations away from the mean.
Let X be a r.v. with expected value <math>\mu</math> and finite variance <math>\sigma^2</math>. Then for any real number? <math>k > 0</math>,
Only the cases <math>k > 1</math> provide useful information. This can be equivalently stated as
