Discrete vs.Continuous random variable, and their distributions

Discrete random variable
The possible value is a countable number, such as coin flip or rolling dice.

Continuous random variable
The possible value is an uncountable (infinite) number, such as asset returns or temperatures.

Discrete distribution
No. of possible outcomes are countable

• If x cannot occur, P(x) =0
• If x can occur, P(x) >0

Discrete Uniform Random Variable

• Probability of all possible outcomes are equal p(x)=1/n
  Examples: Coin flip, die

Binomial Random Variable

• A binary variable that takes on one of two values, usually 1 for success or 0 for failure. Example: coin flip

Binomial probability

• Calcualte the no. of ways to choose x “success” from n-independent trials
  p(x)=P(X=x)={n!/(n-x)!x!}p^x(1-p)^n-x
  where: p is the probability of success on any one trial

Expected value and variance of a binomial random variable

• Expected value = np for n trial, i.e. p for a single trial
• Variance = np(1-p) for n trials. I.e. p (1- p) for single trial,

Continuous distribution:

• No. of possible outcomes are uncountable.

Note:

• Even x can occur, P(x)=0. Therefore, only meaningful to consider P(x1≦X≦x2) rather than a specific value.

Continuous uniform distribution

• X is a random variable that has equal probabilities for taking on values in the interval [a,b]
• f(x) = 1/(b-a), if a≦x≦b
• f(x) = 0, if xb

Continuous Uniform Cumulative Distribution:

• F(x) = 0 if x≦a
• F(x) = (x-a)/(b-a), if a≦x≦b
• F(x) = 1 if x≧b
• For all a≦x1≦x2≦b, then P(X or X>b)=0 and P(X≧b)=1