Expected Value Calculator Formula

Understand the math behind the expected value calculator. Each variable explained with a worked example.

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Formulas Used

Expected Value E(X)

expected_value = ev

Variance

variance_val = ev2 - pow(ev, 2)

Standard Deviation

std_dev = sqrt(abs(ev2 - pow(ev, 2)))

Sum of Probabilities

prob_sum = p1 + p2 + p3

Variables

VariableDescriptionDefault
x1Outcome 110
p1Probability 10.2
x2Outcome 220
p2Probability 20.5
x3Outcome 350
p3Probability 30.3
evDerived value= x1 * p1 + x2 * p2 + x3 * p3calculated
ev2Derived value= pow(x1, 2) * p1 + pow(x2, 2) * p2 + pow(x3, 2) * p3calculated

How It Works

How to Calculate Expected Value

Formula

E(X) = Sum of (xi * pi)

The expected value is the long-run average outcome of a random variable. Multiply each possible outcome by its probability and sum the products. It represents the "center of gravity" of the probability distribution. The variance is E(X^2) - [E(X)]^2.

Worked Example

A game pays $10 with probability 0.2, $20 with probability 0.5, and $50 with probability 0.3.

x1 = 10p1 = 0.2x2 = 20p2 = 0.5x3 = 50p3 = 0.3
  1. 01E(X) = 10*0.2 + 20*0.5 + 50*0.3
  2. 02= 2 + 10 + 15 = 27
  3. 03E(X^2) = 100*0.2 + 400*0.5 + 2500*0.3 = 20 + 200 + 750 = 970
  4. 04Var(X) = 970 - 27^2 = 970 - 729 = 241
  5. 05SD = sqrt(241) ≈ 15.5242

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