Find the expected value of the product of variables π΄ and π΅, denoted as πͺ(π΄π΅), when π΄ and π΅ are independent indicator random variables.
To find the expected value of the product of independent indicator random variables π΄ and π΅, denoted as πͺ(π΄π΅), we can use the following steps:
1. Understand what indicator random variables are: An indicator random variable is a random variable that takes on only two possible values, typically 0 or 1. It indicates the occurrence or non-occurrence of an event.
2. Determine the probability distribution of π΄ and π΅: Since π΄ and π΅ are independent indicator random variables, we know that they have a Bernoulli distribution with probability p and q respectively. Here, p represents the probability of event A occurring and q represents the probability of event B occurring.
3. Define the product variable πͺ(π΄π΅): The product variable πͺ(π΄π΅) is given by the product of π΄ and π΅, that is, πͺ(π΄π΅) = π΄ * π΅. It takes on the values of 0 or 1.
4. Calculate the expected value πΌ(πͺ(π΄π΅)): The expected value of a random variable is the weighted average of all possible values it can take on, where the weights are the probabilities of those values. In this case, since πͺ(π΄π΅) can only take on the values 0 or 1, the expected value can be calculated as πΌ(πͺ(π΄π΅)) = (0 * π(πͺ(π΄π΅) = 0)) + (1 * π(πͺ(π΄π΅) = 1)).
5. Use the probabilities of the product variable πͺ(π΄π΅): Since π΄ and π΅ are independent, the probability of π΄ and π΅ both being 1 is equal to the product of their individual probabilities of being 1, that is, π(πͺ(π΄π΅) = 1) = π(π΄ = 1) * π(π΅ = 1) = p * q.
6. Calculate the expected value: Using the probabilities determined in step 5, we can calculate the expected value πΌ(πͺ(π΄π΅)) = (0 * π(πͺ(π΄π΅) = 0)) + (1 * π(πͺ(π΄π΅) = 1)) = 0 * (1 - π(πͺ(π΄π΅) = 1)) + 1 * π(πͺ(π΄π΅) = 1) = 0 * (1 - p * q) + 1 * (p * q) = p * q.
Thus, the expected value of the product of independent indicator random variables π΄ and π΅, denoted as πͺ(π΄π΅), is πΌ(πͺ(π΄π΅)) = p * q.