Indicator variables

01Turn a count into a sum#

An event indicator equals one when its event occurs and zero otherwise. Its expectation is the probability of that event, because $1P(A)+0P(A^c)=P(A)$.

A count of events can be written as a sum of indicators. Linearity of expectation then adds their individual probabilities. It does not require the events to be independent.

02Count expected errors#

Suppose three predictions have error probabilities $0.1,0.2,0.4$. Let $I_j$ indicate an error on prediction $j$. The total error count is $N=I_1+I_2+I_3$.

The observed count is an integer, but its expectation can be fractional. Correlated errors can change the distribution and variance of the count without changing this sum of marginal probabilities.

03Dependence does not block the mean#

Imagine two checks that always fail together, with failure probability $0.3$. Their failure count is either zero or two. Its expectation is $2(0.3)=0.6$, also obtained by adding the two indicator expectations.

You would need dependence information to compute the chance that at least one check fails from marginal probabilities alone. For the expected number of failures, the marginals suffice.

Define the counted event carefully so each indicator adds exactly one when it occurs. Repeated events, overlapping events, and unequal probabilities can all be handled when the intended count is specified clearly.