Raising the bar

Chapter 5. Discrete Random Variables and Probability Distributions

Unit 5.1. Random variable

At some point you may feel that the amount of new information is overwhelming and you forget many facts. Then it's the time to arrange a global repetition. Repeat everything with proofs. Look for those minor or significant details you missed earlier. Make a list of facts in a logical order. This chapter will be more challenging than all the previous taken together, and you will need to be in a perfect shape to conquer it.

Intuitive definition. A random variable is a variable whose values cannot be predicted.

For example, we can associate random variables with the coin and die. On the other hand, to know the value of $latex \sin x$ it suffices to plug the argument $latex x$ in $latex \sin$. $latex \sin x$ or the number of days in a year are not random variables.

Short formal definition. A random variable is a pair: values plus probabilities.

Working definition of a discrete random variable. A discrete random variable is defined by Table 5.1.

Table 5.1. Discrete random variable with $latex n$ values.

Comments. (1) Most of the time, when people talk about random variables, they assume real-valued variables. Thus, in the first column we have real numbers. This is important to be able to use arithmetic operations.

(2) Capital $latex X$ is used for the variable and lower-case $latex x$'s for its values. The complete form of writing $latex {p_i}$ is $latex P(X = {x_i})$.

(3) Without the second column we would have a deterministic variable.

(4) In the table, $latex X$ takes a finite number of values. This is a simplification. In Poisson distribution we'll see an example of a discrete random variable that takes an infinite number of values.

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