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User:Joerite/math - Wikipedia, the free encyclopedia

User:Joerite/math

From Wikipedia, the free encyclopedia

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${\left ( \frac{T_a}{T_b} \right )}^2 ={\left ( \frac{r_a}{r_b} \right )}^3$

$F=G\frac{{m_a}{m_b}}{d^2}$

$T^2=\left ( \frac{4\pi^2}{Gm_s} \right ) r^3$

$v=\sqrt{\frac{Gm_e}{r}}$

$T=2\pi\sqrt{\frac{r^3}{Gm_e}}$

$d=\frac{1}{2}gt^2$

$d=\sqrt{2dg}$

$x\left ( t \right ) = vt \cos \theta$

$y\left ( t \right ) = vt \sin \theta-\frac{1}{2} gt^2$

$time = \frac{2v \sin \theta}{g}$ α

$max =\frac{v^2\sin^2\theta}{2g}$

$\bar v=\frac{\Delta d}{\Delta t}=\frac{d_1-d_0}{t_1-t_0}$

$d = d 0 + v t$

$\bar a=\frac{\Delta v}{\Delta t}=\frac{v_1-v_0}{t_1-t_0}$

$v = v 0 + a t$

$d=d_0+\frac{1}{2}\left ( v+v_0 \right ) t$

$d=d_0+v_0t+\frac{1}{2}at^2$

$v^2=v_0^2+2a\left ( d-d_0 \right )$

[edit] stats

$\int e^x=f(u)^n$

$\bar{x}=\frac{\sum x_i}{n}$

$s_x=\sqrt{\frac{1}{n-1}\sum \left( x_i-\bar{x}^2\right)}$

$s_p=\sqrt{\frac{\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2}{\left(n_1-1\right)+\left(n_2-1\right)}}$

$\hat{y}=b_0+b_1x$

$b_1=\frac{\sum\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)}{\sum\left(x_i-\bar{x}^2\right)}$

$b_0=\bar{y}-b_1\bar{x}$

$r=\frac{1}{n-1}\sum\left ( \frac{x_i-\bar{x}}{s_x}\right )\left (\frac{y_i-\bar{y}}{s_y}\right )$

$b_1=r\frac{s_y}{s_x}$

$s_b1=\frac{\sqrt{\frac{\sum\left ( y_i-\hat{y}\right )^2}{n-2}}}{\sqrt{\sum\left ( x_i-\bar{x}\right )^2}}$

[edit] Probabilty

$P\left ( A \cup B\right )=P\left ( A\right )+P\left ( B\right )-P\left ( A \cap B\right )$

$P\left ( A|B\right )=\frac{P\left ( A \cap B\right )}{P\left (B\right )}$

$E\left ( X\right )=\mu_x=\sum x_i-p_i$

$Var\left ( X\right )=\sigma^2_x=\sum\left ( x_i-=\mu_x\right )^2p_i$

If X has a bionomial distobution with parameters n and p, then :

$P\left ( X=k\right )={n \choose k}p^k\left ( 1-p\right )^{n-k}$

$μ x = n p$

$\sigma_x=\sqrt{np\left ( 1-p\right )}$

$\mu_{\hat{p}}=p$

$\sigma_{\hat{p}}=\sqrt{\frac{p\left (1-p\right )}{n}}$ If mean x is them mean of a random sample of size n from an infinite population with mean mu and standard diviation sigma, then:

$\mu_{\bar{x}}=\mu$

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$

[edit] Inferential Statisitics

$Standardized test statistic: \frac{statiscic-parameter}{\sigma}$

Confidense interval: statisctic (+|-) (critical value)(\sigma of stat)

Sample Mean: $\frac{\sigma}{\sqrt{n}}$ Sample Porpotion $\sqrt{np\left ( 1-p\right )}$

Differecnce of two samples mean: $\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}$

When $σ 1 = σ 2 :$

$\sigma\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}$

Difference of sample porpution: $\sqrt{\frac{p_1\left (1-p_1\right )}{n_1}+\frac{p_2\left (1-p_2\right )}{n_2}}$

When p₁=p₂

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