Discrete Random Variables

Probability mass functions

1. discrete random variable:a discrete random variable X on the probability space is defined to be a mapping such that
   • the image is a countable subset of
2. The word ‘discrete’ here refers to the condition that X takes only countably many values in
3. The (probability) mass function (or pmf):The (probability) mass function (or pmf) of the discrete random variable X is the function p defined by Thus, is the probability that the mapping takes the value .
4. Theorem:Let be a countable set of distinct real numbers, and let be a collection of real numbers satisfying There exists a probability space and a discrete random variable on such that the probability mass function of is given by

Examples

is a positive integer, is a number in , and .

Bernoulli distribution

This is the simplest non-trivial distribution. We say that the discrete random variable has the Bernoulli distribution with parameter if the image of is , so that takes the values 0 and 1 only.

Such a random variable is often called simply a coin toss. There exists such that and the mass function of is given by

Binomial distribution

We say that has the binomial distribution with parameters and if takes values in and Note that gives rise to a mass function satisfying since, by the binomial theorem

Poisson distribution

We say that has the Poisson distribution with parameter if takes values in and Again, this gives rise to a mass function since

Geometric distribution

We say that has the geometric distribution with parameter if takes values in and As before, note that

Negative binomial distribution

We say that has the negative binomial distribution with parameters and if takes values in and As before, note that using the binomial expansion of

Functions of discrete random variables

Let be a discrete random variable on the probability space and let . It is easy to check that is a discrete random variable on also, defined by Simple examples are: If , the mass function of is given by: since there are only countably many non-zero contributions to this sum. Thus, if with , then while if , then

Expectation

• If is a discrete random variable, the expectation of is denoted by and defined by and the expectation of X is often called the expected value or mean of

• Law of the subconscious statistician:If is a discrete random variable and , then whenever this sum converges absolutely.

• properties of expectation:Let be a discrete random variable and let .

  • If and , then .
  • We have that .

• variance(a measure of the degree of dispersion of about its expectation .):The variance of a discrete random variable is defined by ,we get: We may expand the term to obtain: where as before. Thus we obtain the useful formula

Conditional expectation and the partition theorem

• If is a discrete random variable and , the conditional expectation of given is denoted by and defined by whenever this sum converges absolutely.
• (Partition theorem) If is a discrete random variable and is a partition of the sample space such that for each , then whenever this sum converges absolutely.