Let A be an event. Then IA , the indicator random variable...

Let $A$ be an event. Then $I_{A}$ , the indicator random variable of $A$ , equals 1 if $A$ occurs and equals 0 otherwise. Show that the expectation of the indicator random variable of $A$ equals the probability of $A,$ that is, $E\left(I_{A}\right)=p(A)$

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Key Concepts

Indicator Random Variable

An indicator random variable is a simple function used in probability theory that takes the value 1 if a specified event occurs and 0 if it does not. It serves as a bridge between events and numerical analysis, allowing discrete events to be handled algebraically in the context of random variables.

Expectation of a Random Variable

The expectation (or mean) of a random variable is a fundamental concept in probability that represents the long-run average value of the variable after many repetitions of the experiment. It is calculated as the weighted sum of all possible values, with their associated probabilities acting as the weights.

Relationship between Expectation and Event Probability

For an indicator random variable, the expectation directly corresponds to the probability of the event it represents. Since the variable only takes values 0 and 1, the expected value is computed as 0 times the probability of the event not occurring plus 1 times the probability of the event occurring, which simplifies to the probability of the event itself.

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