# 17.3: Appendix C- Data on some common distributions


## Discrete distributions

Indicator function $$X = I_E$$ $$P(X = 1) = P(E) = p$$ $$P(X = 0) = q = 1 - p$$

$$E[X] = p$$ $$\text{Var} [X] = pq$$ $$M_X (s) = q + pe^s$$ $$g_X (s) = q + ps$$

Simple random variable $$X = \sum_{i = 1}^{n} t_i I_{A_i}$$ (a primitive form) $$P(A_i) = p_i$$

$$E[X] = \sum_{i = 1}^{n} t_ip_i$$ $$\text{Var} [X] = \sum_{i = 1}^{n} t_i^2 p_i q_i - 2 \sum_{i < j} t_i t_j p_i p_j$$ $$M_X(s) = \sum_{i = 1}^{n} p_i e^{st_i}$$

Binomial$$(n, p)$$$$X = \sum_{i = 1}^{n} I_{E_i}$$ with $$\{I_{E_i} : 1 \le i \le n\}$$ iid $$P(E_i) = p$$

$$P(X = k) = C(n, k) p^k q^{n - k}$$

$$E[X] = np$$ $$\text{Var} [X] = npq$$ $$M_X (s) = (q + pe^s)^n$$ $$g_X (s) = (q + ps)^n$$

MATLAB: $$P(X = k) = \text{ibinom} (n, p, k)$$ $$P(X \ge k) = \text{cbinom} (n, p, k)$$

Geometric($$p$$)$$P(X = k) = pq^k$$ $$\forall k \ge 0$$

$$E[X] = q/p$$ $$\text{Var} [X] = q/p^2$$ $$M_X (s) = dfrac{p}{1 - qe^s}$$ $$g_X (s) = \dfrac{p}{1- qs}$$

If $$Y - 1$$ ~ geometric $$(p)$$, so that $$P(Y = k) = pq^{k - 1}$$ $$\forall k \ge 1$$, then

$$E[Y] = 1/p$$ $$\text{Var} [X] = q/p^2$$ $$M_Y (s) = \dfrac{pe^s}{1 - qe^s}$$ $$g_Y (s) = \dfrac{ps}{1 - qs}$$

Negative binomial$$(m, p)$$, $$X$$ is the number of failures before the $$m$$th success.

$$P(X = k) = C(m + k - 1, m - 1) p^m q^k$$ $$\forall k \ge 0$$

$$E[X] = mq/p$$ $$\text{Var} [X] = mq/p^2$$ $$M_X (s) = (\dfrac{p}{1 - qe^s})^m$$ $$g_X (s) = (\dfrac{p}{1 - qs})^m$$

For $$Y_m = X_m + m$$, the number of the trial on which $$m$$th success occurs. $$P(Y = k) = C(k - 1, m - 1) p^m q^{k - m}$$ $$\forall k \ge m$$.

$$E[Y] = m/p$$ $$\text{Var} [Y] = mq/p^2$$ $$M_Y(s) = (\dfrac{pe^s}{1 - qe^s})^m$$ $$g_Y (s) = (\dfrac{ps}{1 - qs})^m$$

MATLAB: $$P(Y = k) = \text{nbinom} (m, p, k)$$

Poisson$$(\mu)$$. $$P(X = k) = e^{-\mu} \dfrac{\mu^k}{k!}$$ $$\forall k \ge 0$$

$$E[X] = \mu$$ $$\text{Var}[X] = \mu$$ $$M_X (s) = e^{\mu (e^s - 1)}$$ $$g_X (s) = e^{\mu (s - 1)}$$

MATLAB: $$P(X = k) = \text{ipoisson} (m, k)$$ $$P(X \ge k) = \text{cpoisson} (m, k)$$

## Absolutely continuous distributions

Uniform$$(a, b)$$ $$f_x (t) = \dfrac{1}{b - a}$$ $$a < t < b$$ (zero elsewhere)

$$E[X] = \dfrac{b + a}{2}$$ $$\text{Var} [X] = \dfrac{(b - a)^2}{12}$$ $$M_X (s) = \dfrac{e^{sb} - e^{sa}}{s(b - a)}$$

Symmetric triangular $$(-a, a)$$ $$f_X (t) = \begin{cases} (a + t)/a^2 & -a \le t < 0 \\ (a - t)/a^2 & 0 \le t \le a \end{cases}$$

$$E[X] = 0$$ $$\text{Var} [X] = \dfrac{a^2}{6}$$ $$M_X (s) = \dfrac{e^{as} + e^{-as} - 2}{a^2 s^2} = \dfrac{e^{as} - 1}{as} \cdot \dfrac{1 - e^{-as}}{as}$$

Exponential$$(\lambda)$$$$f_X(t) = \lambda e^{-\lambda t}$$ $$t \ge 0$$

$$E[X] = \dfrac{1}{\lambda}$$ $$\text{Var} [X] = \dfrac{1}{\lambda^2}$$ $$M_X (s) = \dfrac{\lambda}{\lambda - s}$$

Gamma$$(\alpha, \lambda)$$$$f_X(t) = \dfrac{\lambda^{\alpha} t^{\alpha - 1} e^{-\lambda t}}{\Gamma (\alpha)}$$ $$t \ge 0$$

$$E[X] = \dfrac{\alpha}{\lambda}$$ $$\text{Var} [X] = \dfrac{\alpha}{\lambda^2}$$ $$M_X (s) = (\dfrac{\lambda}{\lambda - s})^{\alpha}$$

MATLAB: $$P(X \le t) = \text{gammadbn} (\alpha, \lambda, t)$$

Normal$$N(\mu, \sigma^2)f_X (t) = \dfrac{1}{\sigma \sqrt{2\pi}} \text{exp} (-\dfrac{1}{2} (\dfrac{t - \mu}{\sigma})^2)$$

$$E[X] = \mu$$ $$\text{Var} [X] \sigma^2$$ $$M_X (s) = \text{exp} (\dfrac{\sigma^2 s^2}{2} + \mu s)$$

MATLAB: $$P(X \le t) = \text{gaussian} (\mu, \sigma^2, t)$$

Beta$$(r, s)$$

$$f_X (t) = \dfrac{\Gamma (r + s)}{\Gamma (r) \Gamma (s)} t^{r -1} (1 - t)^{s - 1}$$ $$0 < t < 1$$, $$r > 0$$, $$s > 0$$

$$E[X] = \dfrac{r}{r + s}$$ $$\text{Var} [X] = \dfrac{rs}{(r + s)^2 (r + s + 1)}$$

MATLAB: $$f_X (t) = \text{beta} (r, s, t)$$ $$P(X \le t) = \text{betadbn} (r, s, t)$$

Weibull($$\alpha, \lambda, \nu$$)

$$F_X (t) = 1 - e^{-\lambda (t - \nu)^{\alpha}}$$, $$\alpha > 0, \lambda >0, \nu \ge 0, t \ge \nu$$

$$E[X] = \dfrac{1}{\lambda^{1/\alpha}} \Gamma (1 + 1/\alpha) + \nu$$ $$\text{Var} [X] = \dfrac{1}{\lambda^{2/\alpha}} [\Gamma (1 + 2/\lambda) - \Gamma^2 (1 + 1/\lambda)]$$

MATLAB: ($$\nu = 0$$ only)

$$f_X (t) = \text{weibull} (a, l, t)$$ $$P(X \le t) = \text{weibull} (a, l, t)$$

## Relationship between gamma and Poisson distributions

• If $$X$$ ~ gamma $$(n, \lambda)$$, then $$P(X \le t) = P(Y \ge n)$$ where $$Y$$ ~ Poisson $$(\lambda t)$$.
• If $$Y$$ ~ Poisson $$(\lambda t)$$, then $$P(Y \ge n) = P(X \le t)$$ where $$X$$ ~ gamma $$(n, \lambda)$$.

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