Learning Notes

Useful equations

Taylor Series

\displaystyle f(x) = f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!} (x-a)^2 + \cdots

\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

Bra-kets Notation

\displaystyle \langle m | n \rangle = \delta_{m,n}

\displaystyle \sum_n | n \rangle \langle n| = I

Probabilities

Bayes’ theorem

\displaystyle P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Mean/Expectation value

For x being an event mapped by random variable X
\displaystyle \left<X\right> = \sum_{x \in \Omega} P(x)\cdot X(x)

For any function f that takes x as an input
\displaystyle \left< f[X(x)] \right> = \int_{\Omega} f[X(x)] P(x) dx

For discrete case
\displaystyle \left< X_N \right> = \frac{1}{N} \sum_{n=1}^N X(n)

Corelations

\displaystyle \left<X_1X_2\right> = \int_\Omega X_1(x_1)X_2(x_2)p_1(x_1)p_2(x_2) dx_1 dx_2

Covariance

Variance, squire of S.D.
\displaystyle \text{var}[X] = \left<[X-\left<X\right>]^2\right>

Covariance Matrix
\displaystyle \langle X_i, X_j \rangle =\langle (X_i - \langle X_i\rangle )(X_j -\langle X_j \rangle ) \rangle =  \langle X_i X_j \rangle - \langle X_i \rangle \langle X_j\rangle

Moments

The n-th moments of f(x) about point c
\displaystyle \mu_f (n) = \int_{-\infty}^{\infty} (x-c)^n \cdot f(x) dx

The n-th moment of random variable X, simply replace f(x) with p(x)
\displaystyle \mu_X (n) = \int_{\Omega} X(x)^n \cdot p(x) dx

The n-th central moment of random variable X, the 2-nd central moment is variance
\displaystyle \overline{\mu}_X (n) = \left<[X-\left<X\right>]^n\right>

Domain Transforms

Fourier Transform

Continuous
\displaystyle X(f) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}{\infty} x(t)e^{-i 2\pi ft} dt

\displaystyle x(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}{\infty} X(f)e^{i 2\pi ft} df

Discrete
\displaystyle F(x) = \frac{1}{\sqrt{N}} \sum_n f(n)e^{-i nx}

\displaystyle f(n) = \frac{1}{\sqrt{N}} \sum_x F(x)e^{i nx}

Laplace Transform (Z-Transform)

First Define
\displaystyle s = \sigma + i\omega

Continuous
\displaystyle F(s) = \int_0^{\infty} e^{-st} f(t) dt

\displaystyle f(t) = \frac{1}{2\pi i} \lim_{T\rightarrow \infty}\int_{\gamma - iT}^{\gamma + iT} e^{st} F(s) ds

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