Discrete Random Variables

Discrete Random Variables

1. Probability Distribution

   • a discrete random variable is a variable that can take individual values (usually integers), each with a given probability
   • the complete set of probabilities for all the possible values of X is called the probability distribution of X
   • a discrete uniform distribution is one where all the probabilities are equal
   • some probability distributions may be given as formulae in terms of x
     • here the probability is changed in respect to the value of x

2. Mean or Expected Value

   • to calculate the mean (or E(X)), each possible value of the random variable is multiplied by its probability, and the products are added together
   • the mean value of a random variable X is also called the expected value, or expectation, of X - this is written as E(X)

3. Expectation of a function of a Discrete Random Variable

   • if X is a discrete random variable and a and b are numbers, then the random variable aX + b is called a linear function of X - it is E(aX + b) = aE(X) + b
   • if g(X) is a function of a discrete random variable X, then: E[g(X)] is the sum of g(xi)pi
   • if f(x) and g(x) are two functions of X, then: E[f(X) + g(X)] = E[f(X)] + E[g(X)]
   • if g(X) is a function of X, then: E[kg(X)] = kE[g(X)]

4. Variance and Standard Deviation

   • the variance of the discrete random variable X is a measure of the spread of the distribution
   • it is as the sum of (xi - u)(xi - u)pi, where u stands for the mean value of X, or E(X)
   • the variance of X is written as Var(X), and is often denoted by lower-case sigma squared
   • standard deviation (lower-case sigma) is calculated by square rooting the variance

5. Variance of a Discrete Random Variable

   • the variance of a linear function aX + b of a random variable X is related to variable of X by the equation Var(aX + b) = a x aVar(X)
   • to find the variance of a non-linear function g(X), use the formula: variance = mean of the square - square of the mean