Expected Value

Expected Value Calculator

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Bernoulli Distribution Expected Value

The expected value (mean) of a Bernoulli random variable X with parameter p is E[X] = p.

\[ E[X] = p \]

Properties:

The expected value represents the long-run average of the random variable
For a Bernoulli distribution, E[X] = p, where p is the probability of success
The expected value is also the "center of mass" of the probability distribution

Understanding Expected Value

Definition & Properties

The Expected Value (or mean) of a random variable X, denoted by E[X], is a measure of the central tendency of the distribution. It represents the long-run average value of the random variable over many independent repetitions of an experiment.

Discrete Random Variables:

\[ E[X] = \sum_{i} x_i \cdot P(X = x_i) \]

where x_i are the possible values of X and P(X = x_i) is the probability of each value.

Continuous Random Variables:

\[ E[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx \]

where f(x) is the probability density function (PDF) of X.

Key Properties:

Linearity: E[aX + b] = a·E[X] + b, for constants a and b
Additivity: E[X + Y] = E[X] + E[Y] for any random variables X and Y
Multiplicativity for independent variables: If X and Y are independent, then E[XY] = E[X]·E[Y]
Law of the Unconscious Statistician: E[g(X)] = ∫ g(x)·f(x) dx for any function g

Interpreting Expected Value

The expected value has several important interpretations:

Long-Run Average:

The expected value represents the arithmetic average of a large number of independent repetitions of the same random experiment. This interpretation is supported by the Law of Large Numbers, which states that the sample mean approaches the expected value as the sample size increases.

Center of Mass:

Physically, the expected value can be thought of as the center of mass (or balancing point) of the probability distribution. If we place weights proportional to the probabilities at each possible value of the random variable along a number line, the expected value is where the line would balance.

Fair Price:

In gambling and decision theory, the expected value represents the "fair price" of a game or decision. If you pay exactly the expected value to play a game, then in the long run, you'll break even (neither win nor lose money on average).

Example: Expected Value in Practice

Consider a simple game where you roll a fair six-sided die and receive a payoff in dollars equal to the number shown.

The expected value is:

\[ E[X] = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6} = \frac{21}{6} = 3.5 \]

This means that if you play this game many times, your average payoff per game will approach $3.50. Therefore, any price below $3.50 to play this game would be favorable to you in the long run.

Expected Values for Common Distributions

Distribution	Parameters	Expected Value	Notes
Bernoulli	p = probability of success	$E[X] = p$	Represents the probability of success
Binomial	n = number of trials p = probability of success	$E[X] = np$	Represents the average number of successes in n trials
Geometric	p = probability of success	$E[X] = \frac{1}{p}$	Average number of trials needed for first success
Poisson	λ = rate parameter	$E[X] = \lambda$	Average number of events in the given interval
Uniform	a = lower bound b = upper bound	$E[X] = \frac{a+b}{2}$	Midpoint of the interval [a,b]
Normal	μ = mean σ = standard deviation	$E[X] = \mu$	Equal to the mean parameter
Exponential	λ = rate parameter	$E[X] = \frac{1}{\lambda}$	Average wait time between events

Applications of Expected Value

Finance & Investment

Expected value is fundamental in portfolio theory and investment decisions. It helps investors calculate the expected return on investments and analyze risk-reward tradeoffs. Insurance companies use expected value to determine premiums based on the probability and magnitude of potential claims.

Game Theory & Decision Making

In game theory and decision analysis, expected value helps determine optimal strategies by weighing potential outcomes by their probabilities. Expected utility theory extends this concept by incorporating risk preferences, allowing for more sophisticated decision-making models under uncertainty.

Engineering & Quality Control

Engineers use expected value to analyze system reliability and performance. In quality control, it helps calculate the expected number of defects or failures, guiding process improvements. Six Sigma methodologies rely heavily on expected value calculations to reduce variability and improve quality.

Medical Decision Making

In medical research and healthcare, expected value helps evaluate treatments by calculating quality-adjusted life years (QALYs) or disability-adjusted life years (DALYs). It's also used in cost-effectiveness analysis to determine the optimal allocation of limited healthcare resources across different interventions.