Definition

1. Noun:
   • The sum of the values of a random variable divided by the number of values: In probability and statistics, the expected value is the long-run average value of repetitions of the experiment it represents. It is a fundamental measure of the center of a probability distribution.

Usage

• The expected value is a key concept for calculating averages in situations involving chance or uncertainty.
• It is used to predict the average outcome of a random process over many trials.
• In decision theory, the expected value can help evaluate the potential payoff of different choices.

Examples

• Noun:
  • The expected value of rolling a fair six-sided die is 3.5.
  • When evaluating an investment, analysts calculate its expected value to assess its potential return.
  • In this game, the expected value of your winnings is zero, meaning it's fair in the long run.

Advanced Usage

• "Law of Large Numbers": This law states that the average of the results obtained from a large number of trials should be close to the expected value.
  • According to the law of large numbers, the casino's profits are stable because they rely on the expected value of each game.
• "Conditional Expected Value": The expected value of a random variable given that a certain condition is or has occurred.
  • We calculated the conditional expected value of the stock price, given that the market had risen that day.

Variants and Related Words

• Expectation (n): Often used synonymously with "expected value" in probability theory.
  • The expectation of the variable X is denoted E[X].
• Mean (n): In probability, the mean of a random variable is its expected value.
  • The population mean is the expected value of a random draw from that population.

Synonyms

• Mathematical Expectation: A formal synonym used in theoretical contexts.
• Average: A more general term; the expected value is the probability-weighted average.
• First Moment: A term used in theoretical statistics, referring to the expected value.

Related Phrases

• Expected Value of a Function: The expected value of a transformation of a random variable.
  • To find the average energy, we compute the expected value of the energy function.
• Expected Monetary Value (EMV): A specific application in business and decision analysis.
  • The project manager used Expected Monetary Value to choose between the two risky proposals.

Related Idioms/Concepts

• "House Edge": In gambling, this represents the casino's average profit per game, which is directly related to the expected value being negative for the player.
  • The house edge in roulette comes from the expected value being less than the player's bet.
• "Fair Game": A game where the expected value of the net gain for a player is zero.
  • A coin toss where you win $1 for heads and lose $1 for tails is a fair game because its expected value is zero.

Noun

1. the sum of the values of a random variable divided by the number of values