Find the expected value of the product of variables 𝐴 and 𝐵, denoted as 𝖪(𝐴𝐵), when 𝐴 and 𝐵 are independent indicator random variables.
To find the expected value of the product of independent indicator random variables 𝐴 and 𝐵, denoted as 𝖪(𝐴𝐵), 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 𝐴 and 𝐵: Since 𝐴 and 𝐵 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 𝖪(𝐴𝐵): The product variable 𝖪(𝐴𝐵) is given by the product of 𝐴 and 𝐵, that is, 𝖪(𝐴𝐵) = 𝐴 * 𝐵. It takes on the values of 0 or 1.
4. Calculate the expected value 𝔼(𝖪(𝐴𝐵)): 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 𝖪(𝐴𝐵) can only take on the values 0 or 1, the expected value can be calculated as 𝔼(𝖪(𝐴𝐵)) = (0 * 𝑃(𝖪(𝐴𝐵) = 0)) + (1 * 𝑃(𝖪(𝐴𝐵) = 1)).
5. Use the probabilities of the product variable 𝖪(𝐴𝐵): Since 𝐴 and 𝐵 are independent, the probability of 𝐴 and 𝐵 both being 1 is equal to the product of their individual probabilities of being 1, that is, 𝑃(𝖪(𝐴𝐵) = 1) = 𝑃(𝐴 = 1) * 𝑃(𝐵 = 1) = p * q.
6. Calculate the expected value: Using the probabilities determined in step 5, we can calculate the expected value 𝔼(𝖪(𝐴𝐵)) = (0 * 𝑃(𝖪(𝐴𝐵) = 0)) + (1 * 𝑃(𝖪(𝐴𝐵) = 1)) = 0 * (1 - 𝑃(𝖪(𝐴𝐵) = 1)) + 1 * 𝑃(𝖪(𝐴𝐵) = 1) = 0 * (1 - p * q) + 1 * (p * q) = p * q.
Thus, the expected value of the product of independent indicator random variables 𝐴 and 𝐵, denoted as 𝖪(𝐴𝐵), is 𝔼(𝖪(𝐴𝐵)) = p * q.