Expected Value
The expected value (or expectation) of a random variable is the probability-weighted average of all possible values it can take. It represents the long-run average outcome if the experiment were repeated many times. Expected value is central to decision theory, gambling, insurance, and financial risk analysis.
Key Formula
E(X) = Σ [x × P(X = x)] for all values x (discrete case)
LaTeX: E(X) = \sum_{x} x \cdot P(X = x) \quad \text{(discrete)}
| Symbol | Meaning | Unit |
|---|---|---|
| E(X) | expected value of the random variable X | same as X |
| x | a possible value of X | same as X |
| P(X = x) | probability that X equals x | unitless |
Worked Example
Problem
A lottery ticket costs ₹10. The prize structure is: ₹100 with probability 0.05, ₹20 with probability 0.15, ₹0 with probability 0.80. What is the expected monetary gain (net of ticket cost)?
Solution
Step 1: Calculate expected prize: E(prize) = 100×0.05 + 20×0.15 + 0×0.80 = 5 + 3 + 0 = ₹8. Step 2: Subtract ticket cost: E(gain) = 8 − 10 = −₹2.
Answer
Expected net gain = −₹2 per ticket; the lottery is unfavorable on average.
Properties of Expected Value
| Property | Formula | Meaning |
|---|---|---|
| Linearity | E(aX + b) = aE(X) + b | Scales and shifts apply directly |
| Sum Rule | E(X + Y) = E(X) + E(Y) | Always holds (any variables) |
| Independence | E(XY) = E(X)E(Y) | Only when X and Y are independent |
| Constant | E(c) = c | Expected value of a constant is itself |
| Non-negative X | E(X) ≥ 0 | If X ≥ 0 always, then E(X) ≥ 0 |
Interactive Tools
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Related Terms
Probability Distribution
A probability distribution is a mathematical function that describes the likelihood of each possible outcome of a random variable. It assigns a probability to every possible value or range of values that the variable can take, with all probabilities summing to 1. Probability distributions are foundational in statistics and are used in fields ranging from insurance and finance to physics and machine learning.
Variance (statistics)
Variance measures the average squared deviation of a random variable from its mean, quantifying how spread out the values in a distribution are. A low variance indicates values cluster tightly around the mean; a high variance indicates they are widely dispersed. Variance is the square of the standard deviation and is fundamental to ANOVA, regression analysis, and portfolio theory.
Mean (statistics)
The mean (arithmetic mean) is the sum of all values in a dataset divided by the number of values, and represents the central or typical value. It is the most commonly used measure of central tendency and is sensitive to extreme values (outliers). The mean is used extensively in data analysis, quality control, and as the foundation for more advanced statistical measures such as variance and standard deviation.
Derived from the Latin expectatio (anticipation, hope). The concept was formalised by Christiaan Huygens in De Ratiociniis in Ludo Aleae (1657) and later by Blaise Pascal and Pierre de Fermat in their correspondence on games of chance.