stats.md

Statistics Quick Reference Guide

Chi-squared Distribution

Formula

$X^2(n) = \text{Gamma}(\frac{n}{2}, \frac{1}{2})$ In a sample:

$\frac{(n-1)S^2}{\sigma^2} \sim X^2(n-1)$

Parameters

Parameter	Meaning
n	Degrees of freedom
S²	Sample variance
σ²	Population variance

Student's t-Distribution

Formula

In a sample: $\frac{\bar{X} - \mu}{S/\sqrt{n}} \sim t(n-1)$ $P(T > t_{p,n}) = p$ Due to symmetry: $t_{1-p,n} = -t_{p,n}$

Parameters

Parameter	Meaning
$\bar{X}$	Sample mean
μ	Population mean
S	Sample standard deviation
n	Sample size
p	Probability value

Confidence Intervals

Normal Distribution (n > 30)

Formula

$\text{Interval} = \bar{X} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

where $z_{\alpha/2} = \Phi^{-1}(1-\alpha/2)$

Parameters

Parameter	Meaning
α	Significance level (e.g., 0.05 for 95% CI)
$z_{\alpha/2}$	Critical value = $\Phi^{-1}(1-\alpha/2)$
σ	Population standard deviation

Unknown Variance

Formula

$\text{Interval} = \bar{X} \pm t_{\alpha/2,n-1}\frac{S}{\sqrt{n}}$

Parameters

Parameter	Meaning
$t_{\alpha/2,n-1}$	Critical value from t-distribution
S	Sample standard deviation

Confidence Interval for Variance

Formula

$\frac{(n-1)S^2}{\chi^2_{1-\alpha/2}} < \sigma^2 < \frac{(n-1)S^2}{\chi^2_{\alpha/2}}$

Parameters

Parameter	Meaning
$\chi^2_{\alpha/2}$	Chi-square critical value with n-1 degrees of freedom
S²	Sample variance
n	Sample size

Maximum Likelihood Estimation (MLE)

Formulas

Likelihood function: $L(\theta) = \prod f(x_i;\theta)$ Log-likelihood function: $\ell(\theta) = \log(L(\theta))$

Parameters

Parameter	Meaning
θ	Parameter to be estimated
$f(x_i;\theta)$	Probability density/mass function
$x_i$	Individual observations

Common Steps for MLE

1. Write the likelihood function
2. Take the natural log
3. Find the derivative with respect to θ
4. Set equal to zero and solve
5. Verify it's a maximum using second derivative