What values does the indicator of an event E take for outcomes inside and outside E?

The indicator of an event E is a Bernoulli variable. Its success probability is equal to the probability of the event E occurring.

Conditions: E is an event in the sample space.; The variable is defined as the indicator of E.

Uppercase letters (like $N$) denote the random variable itself, which is a function or rule before observation. Lowercase letters (like $n$) denote a specific realized value or observation of that variable after the experiment is conducted.

Conditions: The context involves probability notation.; Distinguishing between the theoretical variable and empirical data.

A real-valued random variable assigns a number to each sample-space outcome. This distinguishes the function itself from a particular observed value.

Conditions: The variable is real-valued.; The context is a probability space with a defined sample space.

For the event $E={2,4,6}$, the indicator is one on even die outcomes and zero on {1,3,5}. This creates a Bernoulli variable where success corresponds to rolling an even number.

Conditions: The experiment is rolling a standard six-sided die.; The event E is defined as getting an even number (2, 4, or 6).

A discrete counting model is appropriate when the variable represents a count of items within a finite or countable set, such as the number of people in a family. A continuous measurement model is appropriate for idealized quantities that can take any value in an interval, such as time or unrounded monetary amounts.

Conditions: The variable is a count of distinct entities.; The variable is a measurement of a continuous quantity.