Expected Values

Definition

Expected value is a statistical concept that represents the average or central tendency of a probability distribution. It is the weighted average of all possible outcomes, where the weights are the probabilities of each outcome occurring.

5 Must Know Facts For Your Next Test

1. The expected value is calculated by multiplying each possible outcome by its corresponding probability and then summing the products.
2. Expected values are used to make decisions under uncertainty, as they provide a way to quantify the average or expected outcome of a probability distribution.
3. The expected value of a discrete random variable is the sum of the products of each possible value and its corresponding probability.
4. For a continuous random variable, the expected value is calculated as the integral of the product of the variable and its probability density function over the entire range of the variable.
5. Expected values are an important concept in decision theory, risk analysis, and portfolio management, as they help quantify the potential outcomes and their likelihoods.

Review Questions

• Explain how the expected value is calculated for a discrete probability distribution.
  • For a discrete probability distribution, the expected value is calculated by multiplying each possible outcome by its corresponding probability and then summing the products. Mathematically, this can be represented as $E[X] = \sum_{i=1}^n x_i \cdot p(x_i)$, where $x_i$ represents the possible outcomes and $p(x_i)$ represents the probability of each outcome. This weighted average provides the central tendency or average expected outcome of the probability distribution.
• Describe the relationship between expected value and decision-making under uncertainty.
  • Expected values play a crucial role in decision-making under uncertainty. By calculating the expected value of different possible outcomes, decision-makers can quantify the average or expected outcome of a probability distribution and use this information to make more informed choices. Expected values help decision-makers weigh the potential benefits and risks associated with different alternatives, allowing them to select the option that maximizes the expected value or minimizes the expected cost or risk.
• Analyze how the concept of expected value can be applied in the context of portfolio management.
  • In portfolio management, expected value is used to evaluate the potential returns and risks of different investment options. By calculating the expected return of each asset or investment, portfolio managers can construct a diversified portfolio that maximizes the expected return for a given level of risk. The expected value also helps portfolio managers assess the potential downside risk and make decisions to optimize the risk-return tradeoff. Additionally, the concept of expected value is used in portfolio optimization techniques, such as mean-variance analysis, to determine the optimal allocation of assets within a portfolio.

Related terms

Probability Distribution: A probability distribution is a mathematical function that describes the likelihood of different possible outcomes in a given situation or experiment.

Weighted Average: A weighted average is a calculation that takes into account the relative importance or significance of each item in a data set, rather than treating all items equally.

Central Tendency: Central tendency is a statistical measure that identifies the central or typical value in a dataset, such as the mean, median, or mode.