Mean of a random variable
\[E(X)=\sum x_ip_i,\qquad \sum p_i=1\]
Probability
Random Variables and Probability Distributions
Understand
A random variable \(X\) assigns a real number to each outcome. Its probability distribution lists the values \(x_i\) with probabilities \(p_i=P(X=x_i)\), where every \(p_i\ge0\) and \(\sum p_i=1\).
\[E(X)=\mu=\sum x_ip_i\]
The mean (expectation) is the long-run average value of \(X\).
Example. Toss three coins; let \(X\) be the number of heads. Values 0, 1, 2, 3 with probabilities \(\frac18,\frac38,\frac38,\frac18\); \(E(X)=\frac{0+3+6+3}{8}=\frac32\).
Key Concepts
- List all values of X.
- Find each probability; check they sum to 1.
- E(X) = Σ xᵢpᵢ.
Formula Bank
Solved Examples
Mean score
A spinner shows 1, 2 or 5 with probabilities 1/2, 1/3, 1/6. Find the mean score.
\(E(X)=\frac12+\frac23+\frac56=\frac{3+4+5}{6}=2\).
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A distribution is a table of values and probabilities; the mean is their weighted average.
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