Resources | Subject Notes | Mathematics
This section covers discrete random variables, their probability distributions, the concept of expected value (mean), and variance. These are fundamental concepts in probability and statistics.
A discrete random variable is a variable whose value can only take on a finite number of values or a countably infinite number of values. Examples include the number of heads in three coin flips or the number of defective items in a sample.
A probability distribution describes the probability of each possible value of a discrete random variable. This is often represented by a probability mass function (PMF).
The probability mass function (PMF), denoted by $p(x)$, gives the probability that the random variable $X$ takes on the value $x$.
Key properties of a PMF:
Example: Consider a random variable $X$ representing the number of heads in two coin flips. The possible values for $X$ are 0, 1, and 2. The probability distribution is:
| $x$ | $p(x)$ |
|---|---|
| 0 | $\frac{1}{4}$ |
| 1 | $\frac{2}{4} = \frac{1}{2}$ |
| 2 | $\frac{1}{4}$ |
The expected value (or mean), denoted by $E(X)$ or $\mu$, represents the average value of a discrete random variable over many trials. It is a weighted average of all possible values, where the weights are the probabilities of those values.
The formula for the expected value is:
$$E(X) = \sum_{x} x \cdot p(x)$$Example: Using the coin flip example above:
$$E(X) = 0 \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} + 2 \cdot \frac{1}{4} = 0 + \frac{1}{2} + \frac{1}{2} = 1$$This means that on average, we expect to get 1 head in two coin flips.
The variance, denoted by $Var(X)$ or $\sigma^2$, measures the spread or dispersion of a discrete random variable around its expected value. It quantifies how much the values of the random variable typically deviate from the mean.
The formula for the variance is:
$$Var(X) = E[(X - E(X))^2] = \sum_{x} (x - E(X))^2 \cdot p(x)$$Example: Using the coin flip example and the calculated expected value of 1:
$$Var(X) = (0 - 1)^2 \cdot \frac{1}{4} + (1 - 1)^2 \cdot \frac{1}{2} + (2 - 1)^2 \cdot \frac{1}{4}$$ $$Var(X) = (-1)^2 \cdot \frac{1}{4} + (0)^2 \cdot \frac{1}{2} + (1)^2 \cdot \frac{1}{4}$$ $$Var(X) = \frac{1}{4} + 0 + \frac{1}{4} = \frac{1}{2}$$The variance is often expressed as the standard deviation, which is the square root of the variance:
$$\sigma = \sqrt{Var(X)}$$In our example, the standard deviation is $\sigma = \sqrt{\frac{1}{2}} \approx 0.707$
Understanding discrete random variables, their probability distributions, expected value, and variance is crucial for analyzing and making inferences about data that can only take on a finite or countably infinite number of values. These concepts form the foundation for more advanced statistical analysis.