Final answer:
The expected value of an indicator random variable with an infinite number of outcomes is calculated by multiplying each possible outcome by its probability and summing these products. For an indicator random variable which only takes 0 or 1, the expected value is simply the probability of the event it indicates.
Step-by-step explanation:
Indicator random variables take only the values 0 or 1, representing the occurrence or non-occurrence of an event. For an indicator random variable with an infinite number of outcomes, the expected value, or mean (μ), can still be calculated using the definition of expected value.
This is done by summing the product of each outcome's value and its probability. Specifically, for indicator random variable X, E(X) = μ = Σ xP(x), where the sum is taken over all possible outcomes. If X is an indicator variable, then it only takes the value 1 with probability p (the event occurs) and 0 with probability q = 1 - p (the event does not occur). Hence, E(X) = 0*(1-p) + 1*p = p.
One common example is flipping a fair coin indefinitely. Here, if X represents getting a head on a single flip, the expected value of X would be the probability of getting a head, which is 0.5.
This approach to computing expected value holds regardless of whether the probability distribution is finite or infinite, so long as the series converges absolutely. For complex distributions or those with an infinite number of outcomes, more advanced mathematical techniques like integration for continuous random variables might be necessary to find the expected value.