expected value

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expected value

A student calculates the expected value of rolling a fair six-sided die.

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.
expected value

A student calculates the expected value of rolling a fair six-sided die.

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

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