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Random Variables and Expectation

A random variable attaches a number to every outcome. Its expected value is where the distribution balances; its variance says how far the weight sits from that point.

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A fair die is a random variable X taking the values 1 to 6, each with probability 1/6. Picture each probability as a weight standing on a beam at its value.

E[X] = \sum x\,p(x) = 1(0.167) + 2(0.167) + 3(0.167) + 4(0.167) + 5(0.167) + 6(0.167) = \mathbf{3.5} \\[6pt] \operatorname{Var}(X) = \sum (x - \mu)^2 p(x) = \mathbf{2.917}, \quad \sigma = \sqrt{2.917} = 1.708

Each term x·p(x) is a weight times its distance from 0 — the beam's turning force. The balance point is where the forces on the two sides cancel.

Random Variables and Expectation

A random variable attaches a number to every outcome. Its expected value is where the distribution balances; its variance says how far the weight sits from that point.

Step 1 of 14

Speed

A fair die is a random variable X taking the values 1 to 6, each with probability 1/6. Picture each probability as a weight standing on a beam at its value.

E[X] = \sum x\,p(x) = 1(0.167) + 2(0.167) + 3(0.167) + 4(0.167) + 5(0.167) + 6(0.167) = \mathbf{3.5} \\[6pt] \operatorname{Var}(X) = \sum (x - \mu)^2 p(x) = \mathbf{2.917}, \quad \sigma = \sqrt{2.917} = 1.708

Each term x·p(x) is a weight times its distance from 0 — the beam's turning force. The balance point is where the forces on the two sides cancel.

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Random Variables and Expectation

See also

Random Variables and Expectation

See also