Expected Value

Definition

Expected value is a statistical concept that represents the average or typical outcome of a random event or process. It is calculated by multiplying each possible outcome by its probability and then summing these values to obtain the expected or average result.

5 Must Know Facts For Your Next Test

1. Expected value is a useful tool for decision-making, as it allows individuals or organizations to anticipate the average or typical outcome of a random event or process.
2. The expected value of a discrete random variable is calculated by summing the product of each possible outcome and its corresponding probability.
3. In the context of probability, expected value represents the long-term average or central tendency of a random variable, rather than a specific outcome.
4. Expected value can be used to compare the relative merits of different options or alternatives, as it provides a quantitative measure of the typical or average outcome.
5. The concept of expected value is closely related to the idea of a weighted average, where each value is multiplied by a factor that represents its relative importance or contribution to the overall result.

Review Questions

• Explain how expected value is calculated for a discrete random variable.
  • To calculate the expected value of a discrete random variable, you multiply each possible outcome by its corresponding probability, and then sum these products. Mathematically, this can be expressed as: E(X) = Σ x_i * P(x_i), where x_i represents the possible outcomes and P(x_i) represents the probability of each outcome. This gives you the average or typical value that you would expect to see over many repetitions of the random process.
• Describe how expected value can be used to compare the relative merits of different options or alternatives.
  • Expected value provides a quantitative measure of the typical or average outcome of a random event or process. By calculating the expected value for different options or alternatives, you can compare their relative merits and make more informed decisions. For example, if you are considering investing in two different stocks, you could calculate the expected value of the returns for each stock based on the probabilities of different outcomes. This would allow you to compare the typical or average return you could expect from each investment and make a more informed decision about which one to choose.
• Analyze how the concept of expected value is related to the idea of a weighted average.
  • The concept of expected value is closely related to the idea of a weighted average, as they both involve multiplying each value by a factor that represents its relative importance or contribution to the overall result. In the case of expected value, the 'weights' are the probabilities of each possible outcome. Just as a weighted average gives more importance to values with higher weights, the expected value of a random variable gives more importance to outcomes with higher probabilities. This connection between expected value and weighted averages helps to illustrate how expected value provides a quantitative measure of the typical or average outcome of a random process.

Related terms

Probability: The likelihood or chance that a particular event or outcome will occur, usually expressed as a number between 0 and 1.

Random Variable: A variable whose value is determined by chance or randomness, such as the outcome of a coin flip or the number of defective items in a production batch.

Weighted Average: A type of average where each value is multiplied by a factor that represents its relative importance or contribution to the overall result.