Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a Poisson distribution with mean \(\theta\). Find the conditional expectation \(E\left(X_{1}+2 X_{2}+3 X_{3} \mid \sum_{1}^{n} X_{i}\right)\).

Short answer

The conditional expectation \(E\left(X_{1}+2 X_{2}+3 X_{3} \mid \sum_{1}^{n} X_{i}\right)\) is \(6\theta/n\).

Step-by-step solution

Write down the expression for which the expected value needs to be calculated

The expression is \(X_{1}+2 X_{2}+3 X_{3}\). This is the value whose expected value we are aiming to calculate given the condition.

Formulate the sum of the random variables

The problem gives the condition as \(\sum_{1}^{n} X_{i}\). These are the random variables representing the total sum.

Apply the concept of conditional expectation

The expectation of a function given a particular event can be computed as the function of random variables multiplied by the conditional probability of the event. However, in this case, we recognize that each \(X_i\) in the summation \(\sum_{1}^{n} X_{i}\) is independently and identically distributed so it contributes equally to the total mean. Hence, proportional to their weights, the conditional expectation of the given expression is simply \(\frac{1}{n}E\left[X_{1}+2 X_{2}+3 X_{3}\right]\).

Use properties of expectation and mean of Poisson distribution

Each \(X_i\) has its own contribution, i.e. 1 from \(X_1\), 2 from \(X_2\) , and 3 from \(X_3\). Also, the expectation of \(X_i\) is \(\theta\) since \(X_i\) follows the Poisson distribution. Hence, the expected value becomes then \((1+2+3)\theta/n = 6\theta/n\).