Statistics for Data Science II (BSMA1004) formula sheet

$f_{XY}(t_1, t_2) = P(X = t_1,\ Y = t_2)$

$\sum_{t_1 \in T_x} \sum_{t_2 \in T_y} f_{XY}(t_1, t_2) = 1$

$f_X(t) = \sum_{t' \in T_Y} f_{XY}(t, t')$

$f_Y(t) = \sum_{t' \in T_X} f_{XY}(t', t)$

$f_{Y|X=t}(t') = \frac{f_{XY}(t, t')}{f_X(t)}$

$f_{XY}(t, t') = f_{Y|X=t}(t') \cdot f_X(t)$

$\sum_{t'} f_{Y|X=t}(t') = 1$

$f_{XY}(t, t') = f_X(t) \cdot f_{Y|X}(t' | X = t)$

$f_X(k) = e^{-\lambda/2} \frac{(\lambda/2)^k}{k!}$

$f_{X_1, X_2, \ldots, X_n}(t_1, t_2, \ldots, t_n) = P(X_1 = t_1, X_2 = t_2, \ldots, X_n = t_n)$

$f_{X_1}(t) = \sum_{t_2' \in \mathcal{T}_{X_2}} \sum_{t_3' \in \mathcal{T}_{X_3}} \cdots \sum_{t_n' \in \mathcal{T}_{X_n}} f_{X_1, X_2, \ldots, X_n}(t, t_2', t_3', \ldots, t_n')$

$f_{X_1 X_2}(t_1, t_2) = \sum_{t_3' \in \mathcal{T}_{X_3}} f_{X_1, X_2, X_3}(t_1, t_2, t_3')$

$f_{X_1 | X_2 = 0}(t_1) = \frac{f_{X_1, X_2}(t_1, 0)}{f_{X_2}(0)}$

$f_{XY}(t_1, t_2) = f_X(t_1) \times f_Y(t_2) \quad \text{for all } t_1, t_2$

$f_{X_1,\ldots,X_n}(t_1,\ldots,t_n) = f_{X_1}(t_1) \cdot f_{X_2}(t_2) \cdots f_{X_n}(t_n)$

$f_{X_1,\ldots,X_n}(t_1,\ldots,t_n) = \prod_{i=1}^{n} f_X(t_i) = [f_X(t)]^n \text{ (when all } t_i = t\text{)}$

$P(A \cap B) = 1 - \left[2\left(\frac{15}{16}\right)^n - \left(\frac{14}{16}\right)^n\right]$

$P(X > m+n \mid X > m) = P(X > n)$

$P(Y = y) = P\bigl(X = f^{-1}(y)\bigr)$

$P(Y = y_0) = \sum_{i:\, f(x_i) = y_0} P(X = x_i)$

$P(Z = z) = \sum_{\{(x,y) \, :\, g(x,y) = z\}} f_{X,Y}(x,y)$

$f_Y(t) = \sum_{g(t_1, \ldots, t_n) = t} f_{X_1, \ldots, X_n}(t_1, \ldots, t_n)$

$f_Z(z) = \sum_{x=-\infty}^{\infty} f_{XY}(x, z-x)$

$f_Z(z) = \sum_{x=-\infty}^{\infty} f_X(x) f_Y(z-x)$

$\sum_{i=1}^{n} X_i \sim \text{Binomial}(n, p)$

$X + Y \sim \text{Poisson}(\lambda_1 + \lambda_2)$

$F_Z(z) = F_X(z) \cdot F_Y(z)$

$E[X] = \sum_{t \in T_x} t \cdot f_x(t)$

$E[X] = \frac{a+b}{2}$

$E[X] = \frac{1}{p} \quad \text{for geometric}(p)$

$E[X] = \lambda \quad \text{for Poisson}(\lambda)$

$E[c] = c$

$E[g(X)] = \sum_{t} g(t) \cdot P(X = t)$

$E[aX + bY] = a \cdot E[X] + b \cdot E[Y]$

$E[Y] = np$

$E[\text{gain}] = 0 \quad \text{when } a = \tfrac{12}{5},\; b = 6$

$E[\text{gain}] \approx -0.5 \quad \text{when } a = 1,\; b = 4$

$E[X] = np \quad \text{for } X \sim \text{Binomial}(n, p)$

$E[\text{empty bins}] = n\left(1 - \frac{1}{n}\right)^m \approx ne^{-m/n}$

$\text{Var}(X) = E[(X - E[X])^2]$

$\text{SD}(X) = \sqrt{\text{Var}(X)}$

$\text{Var}(aX) = a^2 \text{Var}(X)$

$\text{Var}(X + a) = \text{Var}(X)$

$\text{Var}(X) = E[X^2] - (E[X])^2$

$\text{Cov}(X, Y) = E\left[(X - E[X])(Y - E[Y])\right]$

$\text{Cov}(X, Y) = E[XY] - E[X]\,E[Y]$

$\rho_{XY} = \frac{\text{Cov}(X,Y)}{\sigma_X \, \sigma_Y}$

$-\sigma_X \, \sigma_Y \leq \text{Cov}(X,Y) \leq \sigma_X \, \sigma_Y$

$-1 \leq \rho_{XY} \leq 1$

$P(X \geq c) \leq \frac{\mu}{c}$

$P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}$

$F_X(x) = P(X \leq x)$

$P(a < X \leq b) = F_X(b) - F_X(a)$

$F: \mathbb{R} \to [0,1]$

$F(x) = \frac{1}{1 + \exp\left(-\frac{x - 60}{\sqrt{24}}\right)}$

$P(a < X \leq b) = F(b) - F(a)$

$P(X = x_1) = F(x_1) - F(x_1^-)$

$X \text{ is continuous} \iff F(x) \text{ is continuous for all } x$

$\int_a^b f(x)\,dx = F(b) - F(a)$

$f_X(x) = \frac{d}{dx}F_X(x)$

$F_X(x_0) = \int_{-\infty}^{x_0} f_X(x)\,dx$

$\int_{-\infty}^{\infty} f_X(x)\,dx = 1$

$\int_{\mathrm{support}(X)} f_X(x)\,dx = 1$

$P(A) = \int_A f_X(x)\,dx$

$f_X(x) = \begin{cases} \frac{1}{b-a} & \text{if } a < x < b \\ 0 & \text{otherwise} \end{cases}$

$F_X(x) = \begin{cases} 0 & \text{if } x < a \\ \frac{x-a}{b-a} & \text{if } a \leq x \leq b \\ 1 & \text{if } x > b \end{cases}$

$f_X(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x > 0 \\ 0 & \text{otherwise} \end{cases}$

$F_X(x) = \int_0^x \lambda e^{-\lambda u} \, du = 1 - e^{-\lambda x}$

$P(X > s + t \mid X > s) = e^{-\lambda t}$

$f_X(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$

$f(x) = \frac{1}{b-a} \quad \text{for } a \le x \le b$

$f(x) = \lambda e^{-\lambda x} \quad \text{for } x \ge 0$