Random Variables and Distributions

Gambler’s Rules
0 i N

  • Difference equation

    • $q=1-p$
    • $p_i = pp_{i+1} + qp_{i-1}$

  • Differential equation An equation that relates one or more functions and their derivatives


Random variables

A function from the sample space S to the rea lline


Bernoulli distribution

  • The Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability $p$ and the value 0 with probability $q = 1 − p$.
    • $P_{x=1} = p$
    • $P_{x=0} = 1-p$

Binomial distribution

  • In n independent Bernoulli trails, e.g. flipping a coin $n$ times, the distribution of success is called Binomial distribution
  • Indicator variables
  • Independent and identically distributed (iid)
  • Probability Math Function (PMF)
    • $P_{(x=k)} = p^k(1-p)^{n-k}$
  • Binomial Distribution vs Normal Distribution:
    • Normal distribution describes continuous data which have a symmetric distribution, with a characteristic ‘bell’ shape.
    • Binomial distribution describes the distribution of binary data from a finite sample. Thus it gives the probability of getting r events out of n trials.