User talk:Mindey/MathNotes: Difference between revisions

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${\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.}$

${\displaystyle C_{n}^{k}={\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}$

${\displaystyle C_{n}^{k}=C_{n}^{n-k}}$

${\displaystyle \int {\frac {1}{x}}dx=ln(x)+C}$

${\displaystyle X\ \sim \ {\mathcal {N}}(\mu ,\,\sigma ^{2})\Leftrightarrow }$

PDF:

${\displaystyle p_{X}(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}},\quad \ x,\mu ,\sigma \in \mathbb {R} ,\sigma >0.}$

CDF:

${\displaystyle F_{X}(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)\quad }$, where ${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt,\quad x\in \mathbb {R} .}$

${\displaystyle X}$ is continuous r.v. ${\displaystyle \Leftrightarrow P\{X=x\}=F_{X}(x)-F_{X}(x-0)=0}$

${\displaystyle X}$ is absolutely continous r.v. ${\displaystyle \Leftrightarrow \exists p:\forall x\in \mathbb {R} \ \ F(x)=\int _{-\infty }^{x}p(t)dt}$, or, in discrete case: ${\displaystyle F(x)=\sum _{x_{i}\leqslant x}p_{i}}$

${\displaystyle \int _{-\infty }^{\infty }e^{-t^{2}}dt={\sqrt {\pi }}}$

If u = u(x), v = v(x), and the differentials du = u '(x) dx and dv = v'(x) dx, then integration by parts states that

${\displaystyle \int u(x)v'(x)\,dx=u(x)v(x)-\int u'(x)v(x)\,dx}$

Liate rule

A rule of thumb proposed by Herbert Kasube of Bradley University advises that whichever function comes first in the following list should be u:^[1]