सांख्यिकी Formulas

$\mu = \frac{1}{N}\sum_{i=1}^N x_i$

$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i$

$\sigma^2 = \frac{1}{N}\sum (x_i - \mu)^2$

$s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2$

$\sigma = \sqrt{\sigma^2}$

$R = x_{\max} - x_{\min}$

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

$P(A \cap B) = P(A) \cdot P(B \mid A)$

$P(B \mid A) = \frac{P(A \cap B)}{P(A)}$

$P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)}$

$P(A \cap B) = P(A) P(B)$

$P(n,r) = \frac{n!}{(n-r)!}$

$C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

$\mu = np$

$\sigma^2 = np(1-p)$

$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\bigl(-\tfrac{(x-\mu)^2}{2\sigma^2}\bigr)$

$z = \frac{x - \mu}{\sigma}$

$Z \sim N(0, 1)$

$P(|X - \mu| < k\sigma):\ 0.68,\ 0.95,\ 0.997$

$SE = \frac{s}{\sqrt{n}}$

$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

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

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

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

$R^2 = r^2$