Discrete Random Variables

In the next three lab assignments, students will review the concept of a random variable and its probability distribution. They will learn how to determine key parameters such as mean, variance, and standard deviation, and compute probabilities for different outcomes.

Discrete Random Variables

A random variable, typically denoted by \(X\), is a variable that takes on a numerical value determined by chance for each outcome of an experiment. A probability distribution provides the probability for each possible value of \(X\), often presented in a table, formula, or graph.

Example: Suppose you flip a coin. If it lands on tails, you win $1; if it lands on heads, you win $0. Here, the random variable \(X\) has two values: 1 and 0. Since the outcomes are equally likely, the probability distribution is:

\(X\)	0	1
\(P(X=x)\)	\(1/2\)	\(1/2\)

This represents a discrete random variable, where the set of possible values is finite or countably infinite (e.g., the number of coin tosses before the first tail).

Parameters of a Probability Distribution

Mean (Expected Value)

The mean is the central, weighted average value of the random variable. It is computed as:

\[ \begin{align} \mu = E[X] = \sum x \cdot P(X=x) \end{align} \]

Variance and Standard Deviation

Variance measures the spread or dispersion of the random variable around the mean. It can be calculated using:

\[ \begin{align} \sigma^2 = \sum \left[(x-\mu)^2 \cdot P(X=x)\right] \end{align} \]

Alternatively, for manual computation, the variance may be expressed as:

\[ \begin{align} \sigma^2 = \sum \left[x^2 \cdot P(X=x)\right] - \mu^2 \end{align} \]

The standard deviation is the square root of the variance.