Mathematics¶

Calculus¶

Calculus

$&f'(x) = \lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} \\ &\int_a^b f(x) dx = \lim_{||\Delta|| \to 0} \sum_{i} f(c_i)\Delta x_i \\ &\frac{d}{dx} g(f(x)) = g'(f(x))f'(x) \\ &\frac{d}{dx} \int_0^{f(x)} g(t) dt = g(f(x))f'(x) \\ &f(b) = f(a) + \int_a^b f'(x) \\ &\frac{d}{dx} \int_0^x f(t) dt = f(x) \\$

Linear Algebra¶

Linear Algebra

$&T \in L(V,W) \\ &T v_i = \lambda_i v_i \\ &det(T) = \prod_{i} \lambda_i \\ &ran(T) = \{Tv \mid v\in V\}. \\ &nul(T) = \{v \in V \mid Tv = 0_W\} \\ &\dim(V) = \dim(nul(T) ) + \dim(ran(T)). \\$

Probability Theory¶

Probability Theory

$&\textsf{P}(A\mid B) = \frac {\textsf{P}(A,B)}{\textsf{P}(B)} \\ &\textsf{P}(A\mid B) = \frac{\textsf{P}(B\mid A)\textsf{P}(A)}{\textsf{P}(B)} \\ &A \perp B \iff \textsf{p}(A,B) = \textsf{p}(A) \textsf{p}(B) \\ &\mathbb{E}[X] = \int_{X} x\textsf{p}(x)dx \\ &\textsf{Var}[X] \stackrel{\Delta}{=} \mathbb{E}[X^2]-(\mathbb{E}[X])^2 \\ &\textsf{Cov}[X,Y] \stackrel{\Delta}{=} \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] \\ &\textsf{Cor}(X,Y) \stackrel{\Delta}{=} \textsf{Cov}(X,Y) / \sqrt{\textsf{Var}(X)\textsf{Var}(Y)}$

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