Discrete random variables and expectation

Expectation and variance

For a discrete random variable, the expectation weights each value by its probability:

Pearson Edexcel 9FM0 formulae booklet: included, under Discrete distributions, along with E(g(X)) = Σ g(x)P(X = x). Set the values, probabilities and products out in a table before summing. E(X) is the balance point of the distribution: the value about which Σ (x − μ)P(X = x) is zero. It need not be the most probable value, and it need not be a value the variable can take. A fair die has mean 3.5 and no face shows 3.5. The mode is the most probable outcome.

Functions, scaling and shifting

E(g(X)) applies g to each value before weighting. So E(X²) is Σx²P(X = x), which is the sum the variance formula requires. For a linear function, E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).

Adding b moves the mean by b and leaves the variance unchanged, because it changes every value by the same amount. Multiplying by a multiplies the variance by a², because variance is measured in squared units. Doubling every value therefore quadruples the variance and doubles the standard deviation.

Using mean and variance to assess a model

A mean and a variance can be compared with the values a named model requires. A Poisson distribution has variance equal to its mean. The theoretical binomial variance np(1 − p) is below its mean np for 0 < p < 1. A geometric distribution has variance well above its mean once p is small.

A sample mean and variance can therefore indicate whether a fitted model is plausible, but sampling variation means the comparison alone does not prove or disprove the model. In a comment on the suitability of a model, quote both calculated values, state what the proposed family requires and name the direction of any mismatch.

The median of a table

The median of a discrete variable is found from the cumulative probabilities rather than by solving an equation. Run down the table adding probabilities. Under the lower-median convention, the median is the smallest value m for which F(m) ≥ 0.5. Interpolation does not apply, because the variable takes the listed values and nothing between them.

Define the median convention before using it. When the running total is exactly 0.5 at a value, the lower-median convention gives that value. Under the interval definition, every value from that one up to the next is a median, because at least half the probability lies on each side. The midpoint of the two is a third convention, matching the treatment of a data set with an even number of readings. State which convention you are using, and use the one the question specifies.

The discrete uniform, proved

A discrete uniform distribution gives each of n listed outcomes the same probability 1/n. A fair die is the standard case. The mean and variance follow in one line from the standard series sums.

For the values 1 to n, pull the constant probability out of the expectation: E(X) = (1/n)Σx = (1/n) × n(n + 1)/2. Doing the same with the squares, E(X²) = (1/n) × n(n + 1)(2n + 1)/6, which is (n + 1)(2n + 1)/6. Subtracting the square of the mean and taking a common denominator of 12 leaves (n + 1)[2(2n + 1) − 3(n + 1)]/12, and the bracket collapses to n − 1:

Both closed sums come from the compulsory series work. Pearson Edexcel 9FM0 formulae booklet: not included, so reproduce the derivation. The mean is the midpoint of the list, as the symmetry of the distribution requires, and the variance grows with the square of the number of values.