Cheat Sheet Module 1

Cheat Sheet Module 1#

*** - denotes IMPORTANT

Basic Probability Notations#

  • \(p\) - Probability or Probability Mass Function (pmf, attributes, discrete)

  • \(f\) - Probability Distribution Function (pdf, variables, continuous) ***

  • \(F\) - Cumulative Distribution Function (cdf)

  • \(\Omega\) - Support set (sometimes called domain) (pronounced as “Omega”)

  • \(\omega\) - an element in the support (still pronounced as “omega”)

  • \(\omega \in \Omega \implies\) element omega belongs to the support omega.

  • \(\forall \implies\) reads as and means “for all”

Basic Probability Definitions#

  • \(F(x) = p(\omega \leq x) = \int_{-\infty}^{x}f(\omega)d\omega\) (integral will convert to summation for attributes/pmfs)

  • \(p(a \leq x \leq b) = \int_{a}^{b}f(x)dx\) (integral will convert to summation for attributes/pmfs)

  • \(p(\Omega) = \int_{\Omega} f(x)dx = 1\)

Standard (five) Distributions#

1. Uniform Distribution \(\mathcal{U}(a,b)\)#

  • Variable (continuous)

  • \(a\) - lower limit or bound, \(b\)- upper limit or bound

  • \(f(x) = \frac{1}{b-a} \,\, \forall \,\, a \leq x \leq b\) (\(0\) otherwise)

  • \(F(x) = \begin{cases} 0, \,\, \text{if} \,\, x \leq a \\ \frac{x-a}{b-a}, \,\, \text{if} \,\, a \leq x \leq b \\ 1, \,\, \text{if} \,\, x \geq b \end{cases}\)

2. Normal Distribution \(\mathcal{N}(\mu,\sigma^2)\) ***#

  • Variable (continuous)

  • \(\mu\) - Mean, \(\sigma^2\)- Variance

  • \(f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp(-\frac{(x-\mu)^2}{2 \sigma^2}) \,\, \forall \,\, x \in (-\infty,\infty) \)

  • \(F(x)\) - use standard normal tables using z-score!

  • *** Standard Normal Score or z-score \(z(x) = \frac{x - \mu}{\sigma}\)

  • *** \(\Phi(x)\) - cdf of standard normal distribution \(\mathcal{N}(0,1)\)

  • *** \(F(x) = \Phi(z(x))\)

3. Exponential Distribution \(Exp.Dist.(\lambda)\) (used mostly in reliability engineering)#

  • Variable (continuous)

  • \(f(x) = \lambda \exp(-\lambda x) \,\, \forall \,\, x \in [0,\infty) \)

  • \(F(x) = 1 - \exp(-\lambda x)\)

4. Binomial Distribution \(\mathcal{B}(N,p)\)#

  • Attribute (discrete)

  • \(N\) - Number of Trials, \(p\) - probability of success in one trial

  • \(p(x) = \binom{N}{x} p^{x}(1-p)^{N-x} \,\, \forall \,\, x \in \{0,1,2,..., N\} \)

  • \(F(x) = \sum_{i=0}^{x} \binom{N}{i} p^{i}(1-p)^{N-i}\)

5. Poisson Distribution \(Poisson(\lambda)\)#

  • Attribute (discrete)

  • \(p(x) = \frac{e^{-\lambda} \lambda^x}{x!} \,\, \forall \,\, x \in \{0,1,2,..., \infty\} \)

  • \(F(x) = \sum_{i=0}^{x} \frac{e^{-\lambda} \lambda^x}{x!} \)

Other Distributions (in this course)#

6. Bernoulli distribution \(Bernoulli(p)\)#

  • Attribute (Discrete), \(x \in \{0,1\}\)

  • \(p\) is the probability of success, i.e., probability of getting 1.

  • \((1-p)\) is the probability of failure, i.e., probability of getting 0.

7. Students’ t-distribution \(t(n-1)\)#

  • Variable (continuous), \(x \in (-\infty,\infty)\)

  • \(n\) is the sample size typically

  • \(n-1\) degrees of freedom

8. Chi-squared Distribution \(\chi^2(n-1)\)#

  • Variable (continuous) but can only take positive real numbers \(x \in [0,\infty)\)

  • \(n\) is the sample size typically

  • \(n-1\) degrees of freedom

Sample from a Population Distribution#

  • Each sample has an associated size \(n\)

  • To draw a histogram of the sample, use approximately \(\sqrt{n}\) bins.

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