Expected Value
Expected value, variance and standard deviation of a discrete random variable, Bernoulli formula and binomial distribution with a solution.
The expected value calculator finds the expected value, variance and standard deviation of a discrete random variable from its probability distribution, and solves Bernoulli trial problems: the probability of k successes in n trials, the mean and variance of the binomial distribution and the most probable number of successes.
Discrete random variable
| Characteristic | Formula |
|---|---|
| Expected value | M(X) = E(X) = Σxipi |
| Variance | D(X) = M(X²) − (M(X))² = Σxi²pi − (M(X))² |
| Standard deviation | σ(X) = √D(X) |
The probabilities of all values must add up to 1 — the calculator checks this.
Example. A die: values 1…6 with probability 1/6.
M(X) = (1 + 2 + … + 6) / 6 = 3.5, M(X²) = 91/6, D(X) = 91/6 − 3.5² = 35/12.
Bernoulli trials and the binomial distribution
Pn(k) = Cnk pk qn−k, q = 1 − p; M(X) = np, D(X) = npq
Example. A coin is tossed 10 times. The probability of exactly 5 heads:
P10(5) = C105 · (1/2)10 = 252/1024 ≈ 0.2461.
The most probable number of successes k0 satisfies np − q ≤ k0 ≤ np + p.
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