Statistics II — Formula Sheet
Bachelor’s Degree in Management · Academic Year 2025/2026
Polytechnic University of Lisbon
Probability Models
Discrete Probability Models
Binomial Distribution
\[X \sim \mathrm{Binomial}(n,\, p)\]
\[P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x = 0, 1, \ldots, n\]
\[E(X) = np \qquad V(X) = npq \qquad q = 1-p\]
Hypergeometric Distribution
\[X \sim \mathrm{Hypergeometric}(N,\, M,\, n) \qquad p = \dfrac{M}{N}\]
\[P(X = x) = \dfrac{\displaystyle\binom{M}{x}\binom{N-M}{n-x}}{\displaystyle\binom{N}{n}}, \quad \max(0,\, M+n-N) \leq x \leq \min(M,\, n)\]
\[E(X) = np \qquad V(X) = npq\,\dfrac{N-n}{N-1}\]
Poisson Distribution
\[X \sim \mathrm{Poisson}(\lambda),\quad \lambda > 0\]
\[P(X = x) = \dfrac{e^{-\lambda}\,\lambda^x}{x!}, \quad x = 0, 1, 2, \ldots\]
\[E(X) = V(X) = \lambda\]
Geometric Distribution
\[X \sim \mathrm{Geometric}(p), \quad 0 \leq p \leq 1\]
\[f(x) = p(1-p)^{x-1}, \quad x = 1, 2, \ldots \qquad F(x) = 1 - (1-p)^k, \quad k \leq x < k+1\]
\[E(X) = \dfrac{1}{p} \qquad V(X) = \dfrac{1-p}{p^2}\]
Continuous Probability Models
Uniform Distribution
\[X \sim \mathrm{Uniform}(a,\, b), \quad a,b \in \mathbb{R},\; a < b\]
\[F(x) = \begin{cases} 0 & x < a \\ \dfrac{x-a}{b-a} & a \leq x < b \\ 1 & x \geq b \end{cases}\]
\[E(X) = \dfrac{a+b}{2} \qquad V(X) = \dfrac{(b-a)^2}{12}\]
Exponential Distribution
\[X \sim \mathrm{Exponential}(\lambda), \quad \lambda > 0\]
\[F(x) = \begin{cases} 0 & x \leq 0 \\ 1 - e^{-\lambda x} & x > 0 \end{cases}\]
\[E(X) = \dfrac{1}{\lambda} \qquad V(X) = \dfrac{1}{\lambda^2}\]
Normal Distribution
\[X \sim \mathrm{Normal}(\mu,\, \sigma), \quad \mu \in \mathbb{R},\; \sigma > 0\]
\[E(X) = \mu \qquad V(X) = \sigma^2\]
\[Z = \dfrac{X - \mu}{\sigma} \sim \mathrm{Normal}(0,\,1) \qquad E(Z) = 0 \qquad V(Z) = 1\]
\[\Phi(z) = P(Z \leq z) \qquad \Phi(-z) = 1 - \Phi(z)\]
t-Student Distribution
\[T \sim t_{(n)}, \quad n \text{ degrees of freedom}\]
\[E(T) = 0 \qquad V(T) = \dfrac{n}{n-2},\quad n > 2\]
Chi-Square Distribution
\[Q \sim \chi^2_{(n)}, \quad n \text{ degrees of freedom}\]
\[E(Q) = n \qquad V(Q) = 2n\]
\[Y = \sum_{i=1}^{n} Z_i^2 \sim \chi^2_{(n)}, \quad Z_i \sim \mathrm{Normal}(0,\,1) \text{ i.i.d.}\]
Sampling
Sample Statistics
\[\bar{X} = \dfrac{1}{n}\sum_{i=1}^{n} X_i\]
\[S^2 = \dfrac{1}{n}\sum_{i=1}^{n}(X_i - \bar{X})^2 = \dfrac{1}{n}\sum X_i^2 - \bar{X}^2\]
\[S'^{\,2} = \dfrac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2 = \dfrac{1}{n-1}\sum X_i^2 - \dfrac{n}{n-1}\bar{X}^2\]
\[S^2 = \dfrac{n-1}{n}\,S'^{\,2} \qquad S'^{\,2} = \dfrac{n}{n-1}\,S^2\]
Sampling Distributions and Confidence Intervals for \(\mu\)
| # | Population | Distribution of \(\bar{X}\) | \((1-\alpha)\times 100\%\) Confidence Interval |
|---|---|---|---|
| 1 | \(X\sim N(\mu,\sigma)\); \(\sigma^2\) known | \(Z = \dfrac{\bar{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)\) | \(\left(\bar{x} - z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}},\; \bar{x} + z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\right)\) |
| 2 | \(X\sim N(\mu,\sigma)\); \(\sigma^2\) unknown | \(T = \dfrac{\bar{X}-\mu}{S'/\sqrt{n}}\sim t_{(n-1)}\) | \(\left(\bar{x} - t_{\alpha/2}\dfrac{s'}{\sqrt{n}},\; \bar{x} + t_{\alpha/2}\dfrac{s'}{\sqrt{n}}\right)\) |
| 2b | \(X\sim N(\mu,\sigma)\); \(\sigma^2\) unknown, \(n \geq 30\) | \(Z = \dfrac{\bar{X}-\mu}{S'/\sqrt{n}}\;\dot{\sim}\; N(0,1)\) | \(\left(\bar{x} - z_{\alpha/2}\dfrac{s'}{\sqrt{n}},\; \bar{x} + z_{\alpha/2}\dfrac{s'}{\sqrt{n}}\right)\) (approx.) |
| 3a | \(X\sim{?}\); \(\sigma^2\) known, \(n \geq 30\) | \(Z = \dfrac{\bar{X}-\mu}{\sigma/\sqrt{n}}\;\dot{\sim}\; N(0,1)\) | \(\left(\bar{x} - z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}},\; \bar{x} + z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\right)\) (approx.) |
| 3b | \(X\sim{?}\); \(\sigma^2\) unknown, \(n \geq 30\) | \(Z = \dfrac{\bar{X}-\mu}{S'/\sqrt{n}}\;\dot{\sim}\; N(0,1)\) | \(\left(\bar{x} - z_{\alpha/2}\dfrac{s'}{\sqrt{n}},\; \bar{x} + z_{\alpha/2}\dfrac{s'}{\sqrt{n}}\right)\) (approx.) |
Sampling Distribution and Confidence Interval for \(p\)
| Population | Distribution of \(\hat{p} = X/n\) | \((1-\alpha)\times 100\%\) Confidence Interval |
|---|---|---|
| \(X\sim \mathrm{Binomial}(n,p)\), \(n\geq 30\) | \(\dfrac{\hat{p}-p}{\sqrt{\hat{p}(1-\hat{p})/n}}\;\dot{\sim}\; N(0,1)\) | \(\left(\hat{p} - z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}},\; \hat{p} + z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\right)\) (approx.) |
Sampling Distribution and Confidence Interval for \(\sigma^2\)
| Population | Distribution of \(S'^{\,2}\) | \((1-\alpha)\times 100\%\) Confidence Interval |
|---|---|---|
| \(X\sim N(\mu,\sigma)\) | \(Q = \dfrac{(n-1)S'^{\,2}}{\sigma^2}\sim\chi^2_{(n-1)}\) | \(\left(\dfrac{(n-1)s'^{\,2}}{q_{\sup}},\; \dfrac{(n-1)s'^{\,2}}{q_{\inf}}\right)\) |
\[P(Q > q_{\inf}) = 1 - \dfrac{\alpha}{2} \qquad P(Q > q_{\sup}) = \dfrac{\alpha}{2}\]
Parametric Hypothesis Tests (at level \(\alpha\))
| Hypotheses | Test Statistic (TS) | Critical Region |
|---|---|---|
| \(H_0: \theta = \theta_0 \;\text{vs}\; H_1: \theta \neq \theta_0\) | \(\mathrm{TS}_0\) | \((-\infty,\, q_{\alpha/2}) \cup (q_{1-\alpha/2},\, +\infty)\) |
| \(H_0: \theta \leq \theta_0 \;\text{vs}\; H_1: \theta > \theta_0\) | \(\mathrm{TS}_0\) | \((q_{1-\alpha},\, +\infty)\) |
| \(H_0: \theta \geq \theta_0 \;\text{vs}\; H_1: \theta < \theta_0\) | \(\mathrm{TS}_0\) | \((-\infty,\, q_{\alpha})\) |
\[P(\mathrm{TS} > q_\alpha) = \alpha \qquad \text{Reject } H_0 \text{ if } \alpha > p\text{-value}\]
\[\alpha = P\!\left(\text{Reject } H_0 \mid H_0 \text{ true}\right) \quad \text{(Type I error probability)}\]