Final answer:
To find the expected value of how many new legitimate emails Fred has received, we use the concept of conditional probability. We calculate E(Z) using the law of total expectation, considering the conditional distribution of Y given X. Finally, we substitute the values into the formula to find the expected value.
Step-by-step explanation:
To find the expected value of how many new legitimate emails Fred has received, we need to use the concept of conditional probability. Let's define Z as the number of new legitimate emails. We know that X and Y are independent, so the joint probability distribution of X and Y is the product of their individual probability distributions. The joint distribution can be represented as f(x, y) = P(X = x) * P(Y = y). Given that Fred has received 30 new emails in total, we want to find the expected value of Z, which is E(Z).
Since X and Y are independent, we can use the law of total expectation to calculate E(Z). The law of total expectation states that E(Z) = E(E(Z|X)), where E(Z|X) is the conditional expectation of Z given X. For each value of X, we can evaluate the conditional expectation of Z given X as E(Z|X = x) = x + E(Z|X = x), where E(Z|X = x) is the expected value of Z given X = x. Therefore, we can calculate E(Z) as:
E(Z) = E[E(Z|X)] = E[X + E(Z|X)] = E(X) + E(E(Z|X)).
Since X follows a Poisson distribution with a mean of 10, we know that E(X) = 10. To find E(E(Z|X)), we need to consider the conditional distribution of Y given X. Given X = x, the random variable Y follows a Poisson distribution with a mean of 40. Therefore, E(Z|X = x) is the expected value of the sum of a Poisson distributed random variable with a mean of x and another independent Poisson distributed random variable with a mean of 40-x. This can be calculated using the Poisson sum formula. Finally, we substitute the values into the formula to find E(Z).