Computational Statistics CS 5961/6961


Basic Statistics Definitions

 

  1. Probability density function

    1. Discrete: If X is a discrete random variable with possible values x1, x2, x3, ..., xn, and f (xi ) denotes P ( X = xi ),  the probability that the

      value of X = xi , then   f  is called the (discrete) probability density function (pdf), or simply the density function for X .

      Note that, for all x ,   f (xi ) ≥ 0 and i   f (xi) = 1 , where the elements are summed over all i .

    2. Continuous: If X is a continous random variable, then the (continuous) probability density function is defined as that function f  such that when integrated as below, gives the probabaility that  X  < a , for some constant a,

      cont odf

    3.  integral 1

  2. Cumulative density function

    1. Discrete:

    2. Continuous: cdf

  3. Uniform distribution function

    1. Discrete: A uniform probability space is a finite space in which all the outcomes have the same probability. That is, if X is a discrete random variable with event space {x1, x2, x3, ..., xn}, and f (xi ) denotes P ( X = xi ),  then the probability that the value of X is xi , is simply  1/n .That is,  for all i, 1 ≤ in ,  f (xi ) =  1/n
    2. Continuous: Let X  be a continous random variable and X ∈ [a,b]. Then the (continuous) uniform probability density function f  is defined by

     

    cont unif

     

     

    for

    a  ≤  x < b

    otherwise

     

  4. Binomial distribution function

  5. Normal or Gaussian Distribution

  6. Convolution

    1. Discrete:

    2. Continuous: