Formulas

Statistics for a set of data values `{v_1, v_2, ..., v_n}`

Mean:	`m=1/n sum_(k=1)^n v_k`
Variance:	`var=1/n sum_(k=1)^n (v_k-m)^2`
Standard  deviation:	`sd=sqrt(var)=sqrt(1/n sum_(k=1)^n (v_k-m)^2)`

Continuous Probability

• Exponential Distributions (`0<=t<oo`, `r>0`)
  
  

  Distribution function:	`F text[(] t text[)]=1-e^(-rt)`
  Density function:	`f text[(] t text[)] =r e^(-rt)`
  Mean:	`m=1/r`

• Cauchy Distribution (`-oo<t<oo`)
  
  

  Distribution function:	`F text[(] t text[)] =1/2+1/pi  tan^(-1) t`
  Density function:	`f text[(] t text[)] =1/(pi(1+t^2))`
  Mean:	`m=0`

• Uniform Distributions (`a<=t<=b`)
  
  

  Distribution function:	`F text[(] t text[)] =(t-a)/(b-a)`
  Density function:	`f text[(] t text[)] =1/(b-a)`
  Mean:	`m=(a+b)/2`

• Standard Normal Distribution (`-oo<t<oo`, `c~~0.3989`)
  
  

  Distribution function:	`F text[(] t text[)] =1/2+int_0^t c e^(-s^2text[/]2) ds`
  Density function:	`f text[(] t text[)] =c e^(-t^2text[/]2)`
  Mean:	`m=0`

Error Function

erf` text[(] t text[)] =int_0^t (2 e^(-s^2))/sqrt(pi) ds`

Improper Integrals

`int_a^oo g text[(] t text[)]  dt=lim_(T rarr oo) int_a^T g text[(] t text[)]  dt`

`int_(-oo)^a g text[(] t text[)]  dt=lim_(T rarr -oo) int_T^a g text[(] t text[)]  dt`

`int_(-oo)^oo g text[(] t text[)]  dt=int_(-oo)^0 g text[(] t text[)]  dt+int_0^oo g text[(] t text[)]  dt`