A random variable is said to be discrete if the set of values it can take has either a finite or an infinite but countable number of elements. Its probability distribution can be characterized through a function called probability mass function.
The following is a formal definition.
Definition A random variable is discrete if its support is countable and there exist a function , called probability mass function of , such thatwhere is the probability that will take the value .
A discrete random variable is often said to have a discrete probability distribution.
Let be a random variable that can take only three values (, and ), each with probability . Then, is a discrete random variable. Its support isand its probability mass function is
The Bernoulli distribution, the Binomial distribution and the Poisson distribution are some examples of discrete distributions that are frequently encountered in probability theory and statistics. The first two are examples of distributions having finite support, while the latter is an example of a discrete distribution having an infinite but countable support.
You can read a thorough explanation of discrete random variables in the lecture entitled Random variables.
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