3110 Final Cheat Sheet

\begin{aligned} Change of variables & Distribution function technique \\ F_{Y} (y) & = P (Y \leq y) = P (h (X_{1}, \dots, X_{n}) \leq y), f_{Y} (y) = \frac{\partial F_{Y} (y)}{\partial y} \\ Transformation technique \\ f_{y} (y) & = f_{X} (h^{- 1} (y)) \cdot | \frac{\partial h^{- 1} (y)}{\partial y} | if only one region \\ f_{Y} (y) & = f_{X} (h_{1}^{- 1} (y)) | \frac{d h_{1}^{- 1} (y)}{d y} | + f_{x} (h_{2}^{- 1} (y)) | \frac{| d h_{2}^{- 1} (y)}{d y} | \\ f_{Y_{1}, . . ., Y_{n}} (y_{1}, . . ., y_{n}) & = f_{X_{1}, . . ., X_{n}} (g_{1} (y_{1}, . . ., y_{n}), . . ., g_{n} (y_{1}, . . ., y_{n})) \cdot | J | \\ | J | & = det [\begin{array}{c} \frac{\partial X_{1}}{\partial Y_{1}} & \frac{\partial X_{1}}{\partial Y_{2}} \\ \frac{\partial X_{2}}{\partial Y_{1}} & \frac{\partial X_{2}}{\partial Y_{2}} \end{array}] (integrate the dummy vars) \end{aligned}

\begin{aligned} Order Statistics \\ f_{X_{(1)}} & = n [1 - F_{X} (x)]^{n - 1} f_{X} (x) \\ f_{X_{(n)}} & = n [F_{X} (x)]^{n - 1} f_{X} (x) \\ f_{X_{(r)}} (x) & = \frac{n!}{(r - 1)! (n - r)!} [F_{X} (x)^{r - 1}] f_{X} (x) [1 - F_{X} (x)]^{n - r} \\ f_{\tilde{X}} (x) & = f_{X_{(n + 1)}} (x) = \frac{(2 n + 1)!}{n! \cdot n!} [F_{X} (x)]^{n} f_{X} (x) [1 - F_{X} (x)]^{n} \end{aligned}

\begin{aligned} Unbiasedness \\ b i a s (\hat{θ}) & = E (\hat{θ} - θ) = E (\hat{θ}) - θ \\ M S E (\hat{θ}) & = E [(\hat{θ} - θ)^{2}] = v a r (\hat{θ}) + [E (\hat{θ}) - θ]^{2} \end{aligned}

\begin{aligned} Efficiency (compare unbiased estimators) \\ efficiency (\hat{θ}) & = \frac{C R L B}{V a r (\hat{θ})} \leq 1 if V a r (\hat{θ}) = C R L B \to UMVUE \\ v a r (\hat{θ}) & \geq \frac{1}{n E [{(\frac{\partial \ln f (x; θ)}{\partial θ})}^{2}]} \end{aligned}

Distribution	PDF / PMF	(E(X))	E(X²)	(Var(X))
Uniform(a, b)	$\frac{1}{b - a}, a \leq x \leq b$	$\frac{a + b}{2}$	$\frac{a^{2} + a b + b^{2}}{3}$	$\frac{(b - a)^{2}}{12}$
Beta( $α$ , $β$ )	$\frac{Γ (α + β)}{Γ (α) Γ (β)} x^{α - 1} (1 - x)^{β - 1}, 0 < x < 1$	$\frac{α}{α + β}$	$\frac{α (α + 1)}{(α + β) (α + β + 1)}$	$\frac{α β}{(α + β)^{2} (α + β + 1)}$
Gamma(k, $θ$ )	$\frac{1}{Γ (k) θ^{k}} x^{k - 1} e^{- x / θ}, x > 0$	$k θ$	$k (k + 1) θ^{2}$	$k θ^{2}$
Poisson( $λ$ )	$\frac{λ^{x} e^{- λ}}{x!}, x = 0, 1, 2, \dots$	$λ$	$λ (λ + 1)$	$λ$
Binomial(n, p)	$(\binom{n}{x}) p^{x} (1 - p)^{n - x}, x = 0, 1, \dots, n$	$n p$	$n p (1 - p) + n p^{2}$	$n p (1 - p)$
Geometric(p)	$(1 - p)^{x - 1} p, x = 1, 2, 3, \dots$	$\frac{1}{p}$	$\frac{2 - p}{p^{2}}$	$\frac{1 - p}{p^{2}}$
$χ^{2} (k)$	$\frac{1}{2^{k / 2} Γ (k / 2)} x^{k / 2 - 1} e^{- x / 2}, x > 0$	$k$	$k (k + 2)$	$2 k$
Expntl( $λ$ )	$λ e^{- λ x}, x > 0$	$\frac{1}{λ}$	$\frac{2}{λ^{2}}$	$\frac{1}{λ^{2}}$
Normal( $μ, σ^{2}$ )	$\frac{1}{\sqrt{2 π} σ} e^{- \frac{1}{2 σ^{2}} (x - μ)^{2}}$	$μ$	$μ^{2} + σ^{2}$	$σ^{2}$

Consistency
$\hat{θ}$ is a consistent estimator of the parameter $θ$ if and only if $\exists ϵ > 0$

lim_{n \to \infty} P (| \hat{θ} - θ | < ϵ) = P (- ϵ \leq \hat{θ} - θ \leq ϵ) = P (θ - ϵ \leq \hat{θ} \leq θ + ϵ) = 1

If $\hat{θ}$ is an unbiased estimator of the parameter $θ$ and the $V a r (\hat{θ}) \to 0$ as $n \to \infty$ , then $\hat{θ}$ is a consistent estimator of $θ$

\begin{aligned} Chebyshev’s Theorem \\ P (| X - μ | < k σ) & \geq 1 - \frac{1}{k^{2}}, or P (| X - μ | \geq k σ) \leq \frac{1}{k^{2}} \end{aligned}

Sufficiency
If the conditional given $\hat{θ}$ depends on $θ$ , $\hat{θ}$ is not sufficient

f (X_{1} = x_{1}, \dots, X_{n} = x_{n} | \hat{θ}) = \frac{f (X_{1} = x_{1}, \dots, X_{n} = x_{n}, \hat{θ})}{g (\hat{θ}) \leftarrow pdf of \hat{θ}} = \frac{f (X_{1} = x_{1}, \dots, X_{n} = x_{n})}{g (\hat{θ})}

$\hat{θ}$ is a sufficient estimator iff the joint can be factorized (fact. theorem)

f (X_{1} = x_{1}, \dots, X_{n} = x_{n}; θ) = g (\hat{θ}, θ) \cdot h (x_{1}, \dots, x_{n})

\begin{aligned} Method of Moments (k moments for k parameters) \\ E (X) & = \bar{X}, E (X^{2}) = {\bar{X}}^{2} \\ Method of maximum likelihood estimator \\ L (θ) & = L (θ; x_{1}, \dots, x_{n}) = f (x_{1}, \dots, x_{n}; θ) = \prod_{i = 1}^{n} f (x_{i}; θ) \\ l (θ) & = \ln L (θ) = \ln \prod f (x_{i}; θ) = \sum_{i = 1}^{n} \ln f (x_{i}; θ) \\ \frac{d l (θ)}{d θ} & = 0 to find crit point at \hat{θ} \\ {\frac{d^{2} l (θ)}{d θ^{2}} |}_{θ = \hat{θ}} & = to check maximum, should be < 0 \\ case 2+ parameters: \\ if & [\begin{array}{c} \frac{\partial^{2} ℓ}{\partial θ_{1}^{2}} & \frac{\partial^{2} ℓ}{\partial θ_{1} \partial θ_{2}} \\ \frac{\partial^{2} ℓ}{\partial θ_{2} \partial θ_{1}} & \frac{\partial^{2} ℓ}{\partial θ_{2}^{2}} \end{array}] < 0, then our \hat{θ_{1}}, \hat{θ_{2}} are MLEs of θ_{1}, θ_{2} \end{aligned}

\begin{aligned} Bayesian Estimation \\ (Prior distribution) g (θ) & = Prior belief about θ \\ (Likelihood) L (θ) & = f (x; θ) = Data’s likelihood given θ \\ (Posterior distribution) h (θ | x) & = \frac{f (x, θ)}{f (x)} = \frac{f (x; θ) g (θ)}{f (x)} = \frac{L (θ) g (θ)}{f (x)} \\ L (θ) g (θ) & = Unnormalized posterior \\ f (x) & = \int_{all θ} L (θ) g (θ) d θ = Marginal likelihood \\ h (θ | x) & = \frac{L (θ) g (θ)}{f (x)} = Normalized posterior \\ {\hat{θ}}_{B} & = E (θ | x) = \int_{all θ} θ h (θ | x) d θ = Bayesian estimate \end{aligned}

\sum_{n = 0}^{x} a r^{n} = \frac{a (1 - r^{x + 1})}{1 - r}

\begin{aligned} Confidence Intervals: \\ μ & = \bar{X} \pm z_{1 - α / 2} \cdot \frac{σ}{\sqrt{n}} (known σ) \\ μ & = \bar{X} \pm t_{n - 1, α / 2} \cdot \frac{S}{\sqrt{n}} (unknown σ) \\ μ & = \bar{X} \pm z_{1 - α / 2} \cdot \frac{S}{\sqrt{n}} (CLT) \\ μ_{1} - μ_{2} & = ({\bar{X}}_{1} - {\bar{X}}_{2}) \pm z_{1 - \frac{α}{2}} \cdot \sqrt{\frac{σ_{1}^{2}}{n_{2}} + \frac{σ_{2}^{2}}{n_{2}}} \\ μ_{1} - μ_{2} & = ({\bar{X}}_{1} - {\bar{X}}_{2}) \pm t_{ν, α / 2} \cdot \sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}} (n_{1}, n_{2} \geq 30) \\ μ_{1} - μ_{2} & = ({\bar{X}}_{1} - {\bar{X}}_{2}) \pm t_{n_{1} + n_{2} - 2, α / 2} \cdot \sqrt{S_{p}^{2} (\frac{1}{n_{1}} + \frac{1}{n_{2}})} \\ S_{p}^{2} & = \frac{(n_{1} - 1) S_{1}^{2} + (n_{2} - 1) S_{2}^{2}}{n_{1} + n_{2} - 2} \\ σ^{2} & = (\frac{\sum (X_{i} - μ)^{2}}{χ_{1 - α / 2, n}^{2}}, \frac{\sum (X_{i} - μ)^{2}}{χ_{α / 2, n}^{2}}) (known μ) \\ σ^{2} & = (\frac{(n - 1) S^{2}}{χ_{1 - α / 2, n - 1}^{2}}, \frac{(n - 1) S^{2}}{χ_{α / 2, n - 1}^{2}}) \\ p & = \hat{p} \pm z_{1 - α / 2} \cdot \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} \\ p_{1} - p_{2} & = ({\hat{p}}_{1} - {\hat{p}}_{2}) \pm z_{1 - α / 2} \cdot \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}} \\ \frac{σ_{1}^{2}}{σ_{2}^{2}} & = (\frac{S_{1}^{2}}{S_{2}^{2}} \cdot \frac{1}{F_{1 - α / 2, n_{1} - 1, n_{2} - 1}}, \frac{S_{1}^{2}}{S_{2}^{2}} \cdot \frac{1}{F_{α / 2, n_{1} - 1, n_{2} - 1}}) \end{aligned}

\begin{aligned} Test function: ϕ (x_{1}, \dots, x_{n}) = {\begin{cases} 1 & if (x_{1}, \dots, x_{n}) \in C (reject H_{0}) \\ 0 & otherwise \end{cases} \\ Type I error & = α = P (reject H_{0} ∣ H_{0} true) \\ Type II error & = β = P (fail to reject H_{0} ∣ H_{0} false) \\ Power & = 1 - β = P (reject H_{0} ∣ H_{1} true) \\ π (θ) & = P (reject H_{0} ∣ θ) (power function) \\ Neyman–Pearson Lemma (for simple H_{0} vs simple H_{1}) : \\ L_{0} & = \prod_{i = 1}^{n} f (x_{i}; θ_{0}) (likelihood under H_{0}) \\ L_{1} & = \prod_{i = 1}^{n} f (x_{i}; θ_{1}) (likelihood under H_{1}) \\ \frac{L_{0}}{L_{1}} & \leq k \Rightarrow reject H_{0} \\ Choose k to satisfy P (reject H_{0} ∣ H_{0}) = α \\ Critical region: set of values making \frac{L_{0}}{L_{1}} \leq k \\ Likelihood Ratio Test (general): \\ L (θ) & = \prod_{i = 1}^{n} f (x_{i}; θ) (full likelihood) \\ Λ & = \frac{max L_{0}}{max L} = \frac{L (\tilde{θ})}{L (\hat{θ})} = \frac{\prod_{i = 1}^{n} f (x_{i}, θ_{0})}{\prod_{i = 1}^{n} f (x_{i}, \hat{θ})} \\ Reject H_{0} & if Λ \leq k (or equivalently, - 2 \ln Λ \geq c) \\ Choose k (or c) so that test has level α \\ Asymptotically: - 2 \ln Λ \sim χ_{df}^{2} under H_{0} \end{aligned}

\begin{aligned} Z/T Tests for Mean (t_{n - 1}) \\ C & = {(x_{1}, \dots, x_{n}); | \bar{x} - μ_{0} | \geq Z_{1 - \frac{α}{2}} \frac{σ}{\sqrt{n}}} (2 sided test) \\ θ < θ_{0} : z & \leq - Z_{1 - α}, θ > θ_{0} : z \geq Z_{1 - α} \\ Difference in Means (Unknown Variances) \\ Z_{o b s} & = \frac{\bar{x} - \bar{y} - δ_{0}}{\sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}}} n \geq 30 \\ t & = \frac{\bar{x} - \bar{y} - δ_{0}}{\sqrt{S_{p}^{2} (\frac{1}{n_{1}} + \frac{1}{n_{2}})}} \sim t_{n_{1} + n_{2} - 2} n < 30, pooled σ^{2} \\ Test for Variances \\ Known μ : χ^{2} & = \frac{n {\hat{σ}}^{2}}{σ_{0}^{2}} = \frac{\sum (x_{i} - μ)^{2}}{σ_{0}^{2}} \sim χ_{n}^{2} \\ Unknown μ : χ^{2} & = \frac{(n - 1) s^{2}}{σ_{0}^{2}} \sim χ_{n - 1}^{2} \\ Reject H_{0} & : χ_{o b s}^{2} < χ_{α / 2}^{2} or χ_{o b s}^{2} > χ_{1 - α / 2}^{2} \\ F & = \frac{S_{1}^{2}}{S_{2}^{2}} \sim F_{n_{1} - 1, n_{2} - 1} \\ Reject H_{0} & : F_{o b s} > F_{1 - α / 2}, or F_{o b s} < F_{α / 2} \\ Binomial Proportion Test (Exact) \\ Two-sided & : X \geq K (α / 2) or X \leq K^{'} (α / 2), K smallest and K^{'} largest for \frac{α}{2} \\ One-sided & : X \geq K (α), where P (X \geq K | H_{0}) \leq α \\ Binomial Proportion (Normal Approx.) \\ Z & = \frac{X - n θ_{0}}{\sqrt{n θ_{0} (1 - θ_{0})}} \approx N (0, 1) \\ Continuity correction: \\ Z & = \frac{(x \pm \frac{1}{2}) - n θ_{0}}{\sqrt{n θ_{0} (1 - θ_{0})}} \\ Use + \frac{1}{2} & : x \geq n θ_{0}, - \frac{1}{2} if x < n θ_{0} \\ Chi-squared Test, Reject if χ_{o b s}^{2} \geq χ_{d f, 1 - α}^{2} \\ \sum_{i = 1}^{K} Z_{i}^{2} & \sim χ_{k}^{2} = \sum {(\frac{x_{i} - n_{i} θ_{i}}{\sqrt{n_{i} θ_{i} (1 - θ_{i})}})}^{2} = \sum_{i} \sum_{j} \frac{(f_{i j} - E_{i j})^{2}}{E_{i j}} \sim χ_{d f}^{2} \\ d f & = (k - 1) (c - 1), or k (c - 1) if θ_{j}^{'} s are given \end{aligned}

To test for association (independence):

E_{i j} = n {\hat{π}}_{i j} = n {\hat{π}}_{i \cdot} {\hat{π}}_{\cdot j}, d f = H_{a} - H_{0}

Step 1. Estimate the parameter for the assumed distribution
Step 2. Compute the probability for each observation under the assumed distribution
Step 3. Compute the expected frequencies
Step 4. Test the goodness-of-fit of the assumed distribution to the observed data
$d f = k - 1 -$ # parameters estimated