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 \((n, \lambda)\).