Summary of Statistics Distributions

Summary of Discrete Statistics Distributions

1. Bernoulli Distribution

• Description: A distribution for a single trial with two possible outcomes (success or failure).
• Probability Mass Function (PMF): $$P(X=x)=\{\begin{array}{ll}p& \text{if }x=1\\ 1-p& \text{if }x=0\end{array}$$
• Parameters: $(p\in [0,1])$ (success probability)
• Mean: $(\mu =p)$
• Variance: $({\sigma }^2=p(1-p))$

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2. Binomial Distribution

• Description: The number of successes in ( n ) independent Bernoulli trials.
• PMF: $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k},k=0,1,\dots ,n$$
• Parameters: $(n\in \mathbb{N})$ (number of trials), $(p\in [0,1])$ (success probability)
• Mean (Expected Value): $(\mu =np)$
• Variance: $({\sigma }^2=np(1-p))$

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3. Geometric Distribution

• Description: The number of trials until the first success.
• PMF: $$P(X=k)=(1-p{)}^{k-1}p,k=1,2,\dots$$
• Parameters: $(p\in [0,1])$ (success probability)
• Mean (Expected Value): $(\mu =\frac{1}{p})$
• Variance: $({\sigma }^2=\frac{1-p}{p^2})$

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4. Negative Binomial Distribution

• Description: The number of trials until the $(r)$-th success.
• PMF: $$P(T_k=m)=\left(\genfrac{}{}{0ex}{}{m-1}{k-1}\right)p^k(1-p{)}^{m-k},\text{ for }m\ge k$$
• Parameters: $(k\in \mathbb{N})$ (number of successes), $(p\in [0,1])$
• Mean (Expected Value): $(\mu =\frac{k}{p})$
• Variance: $({\sigma }^2=\frac{k(1-p)}{p^2})$

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5. Poisson Distribution

• Description: Models the number of events occurring in a fixed interval of time/space.
• PMF: $$P(X=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!},k=0,1,2,\dots$$
• Parameters: $(\lambda >0)$ (average rate of occurrence)
• Mean (Expected Value): $(mu=\lambda t)$
• Variance: $({\sigma }^2=\lambda t)$

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Summary of Continuous Statistics Distributions

1. Uniform Distribution

• Description: All intervals of the same length in the range ([a, b]) are equally likely.
• Probability Density Function (PDF): $$f(x)=\{\begin{array}{ll}\frac{1}{b-a},& a\le x\le b\\ 0,& \text{otherwise}\end{array}$$
• Parameters: $(a,b\in \mathbb{R})$, $(a<b)$
• Mean (Expected Value): $(\mu =\frac{a+b}{2})$
• Variance: $({\sigma }^2=\frac{(b-a{)}^2}{12})$

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2. Normal (Gaussian) Distribution

• Description: The bell-shaped curve, symmetric around the mean. This one is ugly, and you’ll be using the Z- Tables instead of this… hopefully.

• PDF:

  $$f(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right),x\in \mathbb{R}$$

• Parameters: $(\mu \in \mathbb{R})$ (mean), $(\sigma >0)$ (standard deviation)

• Mean (Expected Value): $(\mu )$

• Variance: $({\sigma }^2)$

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3. Exponential Distribution

• Description: Models the time between independent events occurring at a constant rate.
• PDF: $$f(x)=\{\begin{array}{ll}\lambda e^{-\lambda x},& x\ge 0\\ 0,& \text{otherwise}\end{array}$$
• Parameters: $(\lambda >0)$ (rate parameter)
• Mean (Expected Value): $(\mu =\frac{1}{\lambda })$
• Variance: $({\sigma }^2=\frac{1}{{\lambda }^2})$

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4. Gamma Distribution

• Description: Generalizes the Exponential distribution; models waiting time for ( k ) events.
• PDF: $$f(x)=\{\begin{array}{ll}\frac{\lambda }{(k-1)!}(\lambda t{)}^{k-1}{e}^{-\lambda x}& \\ 0,& \text{otherwise}\end{array}$$
• PMF

$$F(t)=1-e^{-\lambda t}\sum _{j!}^{k-1}\frac{(\lambda t{)}^j}{j!}\text{ if }k\text{ is a positive integer, and }t\ge 0$$

• Parameters:
  • $(k>0)$: Shape parameter
  • $(\lambda >0)$: Rate parameter
• Mean (Expected Value): $(\mu =\frac{k}{\lambda })$
• Variance: $({\sigma }^2=\frac{k}{{\lambda }^2})$

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5. Beta Distribution

• Description: Models proportions or probabilities in the interval ([0, 1]).
• PDF: $$f(x)=\{\begin{array}{ll}\frac{x^{\alpha -1}(1-x{)}^{\beta -1}}{B(\alpha ,\beta )},& 0\le x\le 1\\ 0,& \text{otherwise}\end{array}$$
• Parameters: $(\alpha ,\beta >0)$
• Mean (Expected Value): $(\mu =\frac{\alpha }{\alpha +\beta })$
• Variance: $({\sigma }^2=\frac{\alpha \beta }{(\alpha +\beta {)}^2(\alpha +\beta +1)})$

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6. Chi-Square Distribution

• Description: The distribution of the sum of squared standard normal variables. This one is ugly, and you’ll be using the Chi-Square - Tables instead of this… hopefully.
• PDF: $$f(x)=\{\begin{array}{ll}\frac{1}{2^{k/2}\Gamma (k/2)}{x}^{(k/2)-1}{e}^{-x/2},& x>0\\ 0,& \text{otherwise}\end{array}$$
• Parameters: $(k>0)$ (degrees of freedom)
• Mean: $(\mu =k)$
• Variance: $({\sigma }^2=2k)$

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7. Student’s t-Distribution

• Description: Used for small sample sizes when estimating a population mean. This one is ugly, and you’ll be using the T- Tables instead of this… hopefully.
• PDF: $$f(x)=\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sqrt{\nu \pi }\Gamma \left(\frac{\nu }{2}\right)}{\left(1+\frac{x^2}{\nu }\right)}^{-\frac{\nu +1}{2}},x\in \mathbb{R}$$
• Parameters: $(\nu >0)$ (degrees of freedom)
• Mean: $(\mu =0)$ (for $(\nu >1))$
• Variance: $({\sigma }^2=\frac{\nu }{\nu -2})$ (for $(\nu >2)$)

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8. Log-Normal Distribution

• Description: Models a variable whose logarithm follows a normal distribution.
• PDF: $$f(x)=\{\begin{array}{ll}\frac{1}{x\sigma \sqrt{2\pi }}\exp \left(-\frac{(\ln x-\mu {)}^2}{2{\sigma }_{l{n}_x}}\right),& x>0\\ 0,& \text{otherwise}\end{array}$$
• Parameters: $(\mu \in \mathbb{R})$ (mean of log), $(\sigma >0)$ (standard deviation of log)
• Mean (Expected Value): $({\mu }_X=e^{\mu +{\sigma }^2/2})$
• Variance: $({\sigma }_X^2=e^{2\mu +{\sigma }^2}(e^{{\sigma }^2}-1))$

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9. Weibull Distribution

• Description: Models failure times and reliability.
• PDF:

$$f(x)=\{\begin{array}{ll}\frac{\beta }{x})(\lambda x{)}^{\beta }{e}^{-(\lambda x{)}^{\beta }},& x\ge 0\\ 0,& \text{otherwise}\end{array}$$

• Parameters:
  • $(k>0)$: Shape parameter
  • $(\lambda >0)$: Scale parameter
• Mean (Expected Value): $(\mu =\lambda \Gamma \left(1+\frac{1}{k}\right))$
• Variance: $({\sigma }^2={\lambda }^2\left[\Gamma \left(1+\frac{2}{k}\right)-{\left(\Gamma \left(1+\frac{1}{k}\right)\right)}^2\right])$