Problem 11 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Let \(X_{1}, X_{2}\), and \(X_{3}\) be three independent Geo \((p)\) distributed random variables, and let \(Z=X_{1}+X_{2}+X_{3}\). a. Show for \(k \geq 3\) that the probability mass function \(p_{Z}\) of \(Z\) is given by $$ p_{Z}(k)=\mathrm{P}\left(X_{1}+X_{2}+X_{3}=k\right)=\frac{1}{2}(k-2)(k-1) p^{3}(1-p)^{k-3} $$ b. Use the fact that \(\sum_{k=3}^{\infty} p_{Z}(k)=1\) to show that $$ p^{2}\left(\mathrm{E}\left[X_{1}^{2}\right]+\mathrm{E}\left[X_{1}\right]\right)=2 $$ c. Use \(\mathrm{E}\left[X_{1}\right]=1 / p\) and part \(\mathbf{b}\) to conclude that $$ \mathrm{E}\left[X_{1}^{2}\right]=\frac{2-p}{p^{2}} \text { and } \operatorname{Var}\left(X_{1}\right)=\frac{1-p}{p^{2}} $$

Step-by-Step Solution

Answer

These are preliminary steps; further steps are needed for a complete solution.

Step 1: Understand the Problem

We need to find the probability mass function (PMF) for a sum of three independent geometric random variables and show it satisfies given distribution properties. Next, we use these properties to find expectations and variances of these variables.

Step 2: Define Geometric PMF

The PMF of a geometric distribution (Geo(p)) for a single trial success probability \( p \) is given by \( P(X = k) = (1-p)^{k-1} p \) for \( k = 1, 2, 3, \dots \).

Step 3: Sum of Independent Geometric Variables

For sum \( Z = X_1 + X_2 + X_3 \), use convolution of PMFs since \( X_1, X_2, X_3 \) are independent identically distributed random variables. The convolution for three such random variables is \( P(Z = k) = \sum_{i=1}^{k-2} \sum_{j=1}^{k-i-1} P(X_1 = i) P(X_2 = j) P(X_3 = k-i-j) \).

Key Concepts

Probability Mass Function

The probability mass function (PMF) is a key concept in understanding discrete random variables, like those in a geometric distribution. The PMF provides the probability that a discrete random variable is exactly equal to a specific value. For a geometric distribution with a success probability of a single trial being \( p \), the PMF is expressed as \( P(X = k) = (1-p)^{k-1} p \). Here, \( k \) represents the number of trials required for the first success.

When you have multiple independent geometric random variables, such as \( X_1, X_2, \) and \( X_3 \), the PMF for their sum, \( Z = X_1 + X_2 + X_3 \), shows the probability of achieving a certain number of successes over these trials. Because they are independent, the PMF of the sum \( Z \) is calculated using the convolution of their individual PMFs.

• This convolution involves summing over all combinations of these variables adding up to the desired result.
• It's derived mathematically by considering each trial's probability and how they combine to form \( Z \).

Expectation of Random Variables

The expectation, or expected value, is a measure of the central tendency for a random variable. It provides a prediction of what you would expect as an average outcome of repeated trials. For a geometric random variable \( X \) with parameter \( p \), the expected value is given by \( \mathrm{E}[X] = \frac{1}{p} \).

This expectation represents the average number of trials needed to get the first success in a sequence of independent trials. When dealing with multiple variables or sums of variables, as in the exercise, the expectation of the sum can be found by summing the expectations of the individual variables:

\[ \mathrm{E}[Z] = \mathrm{E}[X_1] + \mathrm{E}[X_2] + \mathrm{E}[X_3] = \frac{3}{p} \]

• The linearity of expectation allows us to simplify calculations, avoiding complex recalculations when combining sums.
• Understanding expectation helps in assessing what results are typical or central for random processes.

Variance of Random Variables

Variance is a measure of how much the outcomes of a random variable differ from the expected value. For a geometric distribution with parameter \( p \), the variance of a single random variable \( X \) is expressed as \( \operatorname{Var}(X) = \frac{1-p}{p^2} \).

Variance helps in understanding the spread or dispersion in the data. If the variance is high, the data points are spread out over a wider range of values, meaning that there is greater unpredictability in the results. When combining multiple random variables, like in the exercise, the variance of their sum \( Z = X_1 + X_2 + X_3 \) can be found by adding their individual variances:

\[ \operatorname{Var}(Z) = \operatorname{Var}(X_1) + \operatorname{Var}(X_2) + \operatorname{Var}(X_3) = \frac{3(1-p)}{p^2} \]

• Variance is always non-negative, as it measures the average squared deviation from the mean.
• Understanding variance is crucial for evaluating the reliability and uncertainty of predictions based on random variables.