The mean of a random discrete variable, often referred to as the expected value, is a fundamental concept in probability and statistics. It provides a measure of the central tendency of the variable, indicating where most of its values tend to cluster.
To calculate the mean of a discrete random variable, you multiply each possible value of the variable by its corresponding probability and then sum all these products. The formula can be represented as:
Mean (μ) = Σ [x * P(x)]
where:
- x represents each value that the random variable can take.
- P(x) is the probability of each value.
For example, consider a discrete random variable X that represents the roll of a fair six-sided die. The possible values for X are 1, 2, 3, 4, 5, and 6, each with a probability of 1/6. The mean can be calculated as follows:
Mean (μ) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5
This result, 3.5, indicates the average outcome of rolling the die many times. It’s important to note that the mean does not have to be a value that the random variable can take; in this case, 3.5 is not an outcome on a single die roll but reflects an average over many rolls.