User:Joerite/math

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

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

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

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

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

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

$$d=\sqrt{2dg}$$

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

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

$$time = \frac{2v \sin \theta}{g}$$α

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

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

$$d=d_0+vt$$

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

$$v=v_0+at$$

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

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

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

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

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

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

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

$$\hat{y}=b_0+b_1x$$

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

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

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

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

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

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 :

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

$$\mu_x=np$$

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

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

$$\sigma_{\hat{p}}=\sqrt{\frac{p\left (1-p\right )}{n}}$$ 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:

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

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

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 $$\sigma_1=\sigma_2:$$

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

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