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Conditional expectation involves finding the expected value of a random variable given certain conditions or events. In this exercise, we focus on the scenario where the number of visitors at a post office, denoted as the random variable \(X\), is known. Each visitor sends a random number of packages \(Y\) following certain probabilities.

The goal is to find \(E(Y \mid X=x)\), the expected number of packages sent when exactly \(x\) people have visited. Since each person independently mails packages with a specific average (computed from their probabilities), we multiply the constant expected packages per person, \(E(Z) = 2\), by the number of people \(x\). This results in the formula: \(E(Y \mid X=x) = 2x\).

Conditional expectation allows us to predict the average outcome by simplifying it based on known variables.

Conditional variance helps measure how much the number of packages might deviate from their expected number when a known condition is present, such as a specific number of visitors. In this context, we need to determine \(V(Y \mid X=x)\), the variance of the number of packages mailed for a given number of people \(x\).

Just like with conditional expectation, we first compute the variance of packages sent by one individual \(V(Z) = 1\). Since each visitor's acts are independent, variations in package-sending behavior aggregate linearly. Therefore, by multiplying this variance by the number of visitors \(x\), we get \(V(Y \mid X=x) = x\).

Conditional variance helps us understand the potential fluctuations in a result even when a certain aspect is fixed.

The Law of Total Expectation is a powerful tool that allows us to calculate the overall expected value when multiple scenarios or partitions exist, as it aggregates expected values over all possible conditions. In our example, this law helps find the total expected number of packages mailed \(E(Y)\).

Using the conditional expectation derived earlier \(E(Y \mid X)\), we apply the law: \(E(Y) = E[E(Y \mid X)] = E[2X] = 2E(X)\). Since \(E(X) = 20\) for our Poisson-distributed visitors, \(E(Y) = 40\).

By dealing with sub-expectations within each condition and summing them up, this law reveals the complete expected outcome.

The Law of Total Variance breaks down a variance problem considering both within-condition variance and between-condition variance. This law is essential when trying to find \(V(Y)\), the total variance of packages sent by all visitors.

According to this law:

• 1. Calculate \(E[V(Y \mid X)]\), the expected conditional variance, which in our problem is simply \(E(X) = 20\).
• 2. Determine \(V[E(Y \mid X)]\), which reflects the variance across the expectations for each condition. Here, since \(E(Y \mid X) = 2X\), it becomes \(4V(X) = 80\).

Combine these to acquire \(V(Y) = E[V(Y \mid X)] + V[E(Y \mid X)] = 20 + 80 = 100\).
This law ties together within-condition variation and cross-condition variance, offering a comprehensive variance assessment.