Question

If X and Y are independent and identically distributed geometric variables with parameter p, then the moment generating function of (X+Y) is given by

• A. \( \left(\frac{p}{1-qe^t}\right)^2 \)
• B. \( \frac{p}{(1-qe^t)^2} \)
• C. \( \left(\frac{1}{1-qe^t}\right)^2 \)
• D. \( \frac{p}{(1-qe^t)} \)

Hint:

Remember the two main versions of the geometric distribution: one counts trials (\(k=1,2,...\)) and the other counts failures (\(k=0,1,...\)). Their MGFs are different (\(\frac{pe^t}{1-qe^t}\) vs \(\frac{p}{1-qe^t}\)). The options in a multiple-choice question can often give you a clue as to which version is being used.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question asks for the moment generating function (MGF) of the sum of two independent and identically distributed (i.i.d.) geometric random variables. The key property of MGFs is that for independent variables, the MGF of their sum is the product of their individual MGFs.

Step 2: Key Formula or Approach:
1. Identify the MGF of a single geometric random variable. 2. Use the property: If \(X\) and \(Y\) are independent, then \( M_{X+Y}(t) = M_X(t) . M_Y(t) \). 3. The sum of \(r\) i.i.d. geometric variables follows a Negative Binomial distribution. The MGF of a Negative Binomial(\(r, p\)) is \( \left(\frac{p}{1-qe^t}\right)^r \).
Step 3: Detailed Explanation:
First, we need the MGF of a single geometric variable \(X\). The form of the options suggests the version of the geometric distribution that counts the number of failures (\(k=0, 1, 2, \dots\)) before the first success. The MGF for this distribution is: \[ M_X(t) = \frac{p}{1-qe^t} \] where \(q = 1-p\). Since X and Y are i.i.d., they have the same MGF: \[ M_X(t) = M_Y(t) = \frac{p}{1-qe^t} \] Because X and Y are independent, the MGF of their sum, \(Z = X+Y\), is the product of their individual MGFs: \[ M_{X+Y}(t) = M_X(t) . M_Y(t) \] \[ M_{X+Y}(t) = \left(\frac{p}{1-qe^t}\right) . \left(\frac{p}{1-qe^t}\right) \] \[ M_{X+Y}(t) = \left(\frac{p}{1-qe^t}\right)^2 \] This corresponds to the MGF of a Negative Binomial distribution with parameters \(r=2\) and \(p\), which is the expected distribution for the sum of two i.i.d. geometric variables.
Step 4: Final Answer:
The moment generating function of (X+Y) is \( \left(\frac{p}{1-qe^t}\right)^2 \).