To find the expected value of a discrete random variable, you need to follow a systematic approach that involves probability and outcomes. Here’s how to do it:
- Identify the Random Variable: First, determine what your discrete random variable is. This could be something like the number of times a die shows a six in 10 rolls, or the number of customers arriving at a store in an hour.
- List Possible Outcomes: Write down all the possible values that the random variable can take. For example, if you are rolling a die, the possible outcomes are {1, 2, 3, 4, 5, 6}.
- Determine the Probability of Each Outcome: Next, find the probability of each outcome occurring. This might involve using previous knowledge or conducting an experiment. Ensure that the probabilities add up to 1.
- Multiply Outcomes by Their Probabilities: For each possible outcome, multiply the outcome value by its associated probability. This creates a product for each outcome.
- Sum the Products: Finally, add up all the products you calculated in the previous step. This sum gives you the expected value of the random variable.
Formula: The expected value (E) can be calculated using the formula:
E(X) = Σ [x * P(x)]
Here, E(X) is the expected value, x represents the value of the random variable, and P(x) is the probability of the value occurring.
For example, if the random variable X represents the outcome of a fair six-sided die, the expected value can be calculated as follows:
E(X) = (1 * (1/6)) + (2 * (1/6)) + (3 * (1/6)) + (4 * (1/6)) + (5 * (1/6)) + (6 * (1/6)) = 3.5In this case, the expected value tells you the average outcome if you were to roll the die many times. Understanding the expected value can help you make informed decisions based on probability.