Basic Probability Concepts

Moments

- \(r\)th (raw) moment:	\(E[X^r]\)
- \(r\)th central moment:	\(E[(X-E[X])^r]\)
- Coefficient of skewness:	\(\dfrac{E[(X-E[X])^3]}{(\mathrm{Var}[X])^{3/2}}\)
- Coefficient of kurtosis:	\(\dfrac{E[(X-E[X])^4]}{(\mathrm{Var}[X])^{4/2}}\)

Moment Generating Function (MGF)

\[M_X(t)=E[e^{tX}]\] Note that \(M_X^{(k)}(0)=E[X^k]\).

Probability Generating Function (PMF)

\[G_X(t)=E[t^X]\] Note that \(E[X]=G'_X(1)\) and \(\mathrm{Var}[X]=G''_X(1)+G'_X(1)-(G'_X(1))^2\).

Moreover, \(G_X(t)=M_X(\ln(t))\) and \(M_N(t)=G_N(e^t)\).

Model Fitting

1. Visualize raw data, e.g., using histogram and kernel density estimation (KDE).
2. Select candidate distributions based on the shape and characteristics of the data.
3. Estimate parameters , e.g., via maximum likelihood estimation (MLE) and method of moments (MoM).
4. Assess goodness-of-fit, e.g., using Kolmogorov-Smirnov (KS) test, Anderson-Darling test and chi-square test.
5. Model selection, e.g., comparing information criteria (AIC, BIC) and simplicity.

Pearson chi-square goodness-of-fit statistic

\[\chi^2=\dfrac{\sum(O-E)^2}{E}\]