Question

Let \(X_{1}, X_{2}\) be two random variables with joint \(\mathrm{pmf} p\left(x_{1}, x_{2}\right)=(1 / 2)^{x_{1}+x_{2}}\), for \(1 \leq x_{i}<\infty, i=1,2\), where \(x_{1}\) and \(x_{2}\) are integers, zero elsewhere. Determine the joint mgf of \(X_{1}, X_{2}\). Show that \(M\left(t_{1}, t_{2}\right)=M\left(t_{1}, 0\right) M\left(0, t_{2}\right)\).

Short answer

The mgf of \(X_{1}, X_{2}\) is \[M(t_{1}, t_{2}) = \frac{1}{1-\frac{e^{t_{1}}}{2}} \cdot \frac{1}{1-\frac{e^{t_{2}}}{2}}\]. This mgf satisfies \(M(t_{1}, t_{2}) = M(t_{1}, 0) \cdot M(0, t_{2})\).

Step-by-step solution

Definition of mgf

First, remember the definition of the moment generating function (mgf), \(M(t)\) of a random variable \(X\): \[M(t)= E(e^{tX})= \sum (e^{tx} \cdot p(x)) \] for \( t_{1}, t_{2}\) real.

Apply the Definition to the Random Variables

Compute the joint mgf \(M(t_{1}, t_{2})\) of \(X_{1}\) and \(X_{2}\) by applying the definition to the given pmf: \[M(t_{1}, t_{2}) = \sum_{x_{1}=1}^{\infty} \sum_{x_{2}=1}^{\infty} e^{t_{1}x_1 + t_{2}x_{2}} p(x_1, x_2)\] Substitute the given joint pmf to get: \[M(t_{1}, t_{2}) = \sum_{x_{1}=1}^{\infty} \sum_{x_{2}=1}^{\infty} e^{t_{1}x_1 + t_{2}x_{2}} \cdot (1/2)^{x_{1}+x_{2}}\] This simplifies to: \[M(t_{1}, t_{2}) = \sum_{x_{1}=1}^{\infty} \sum_{x_{2}=1}^{\infty} (e^{t_{1}}/2)^{x_{1}} \cdot (e^{t_{2}}/2)^{x_{2}}\]

Simplify the Expression

Can write the sum as a product of two separate sums:\[M(t_{1}, t_{2}) = \left(\sum_{x_{1}=1}^{\infty} (e^{t_{1}}/2)^{x_{1}}\right) \cdot \left(\sum_{x_{2}=1}^{\infty} (e^{t_{2}}/2)^{x_{2}}\right)\]Each of these sums is a geometric series, with the first term of \(e^{t_i}/2\) and the ratio of \(e^{t_i}/2\). Thus, they converge to \[1/(1-e^{t_{i}}/2)\] for \(i =1,2\), assuming \(|e^{t_{i}}/2|<1\). Thus, \[M(t_{1}, t_{2}) = \frac{1}{1-\frac{e^{t_{1}}}{2}} \cdot \frac{1}{1-\frac{e^{t_{2}}}{2}}\]

Show the Product Form

To show the product form, prove that \(M(t_{1}, t_{2}) = M(t_{1}, 0) \cdot M(0, t_{2})\). Notice that when \(t_{2}=0\) in \(M(t_{1}, t_{2})\), \[M(t_{1}, 0) = \frac{1}{1-\frac{e^{t_{1}}}{2}} \cdot \frac{1}{1-\frac{e^{0}}{2}}=\frac{1}{1-\frac{e^{t_{1}}}{2}}\]. Similarly, it holds that \[M(0, t_{2}) = \frac{1}{1-\frac{e^{0}}{2}} \cdot \frac{1}{1-\frac{e^{t_{2}}}{2}}=\frac{1}{1-\frac{e^{t_{2}}}{2}}.\]Therefore, the product form holds as \(M(t_{1}, t_{2}) = M(t_{1}, 0) \cdot M(0, t_{2})\).

Key concepts

Random Variables

In the realm of probability and statistics, random variables are foundational elements that provide a means to quantify outcomes of random phenomena. A random variable, often denoted by capital letters such as X or Y, maps outcomes from a random process to numerical values. For example, consider the roll of a die. The outcome, which can be any number from 1 to 6, can be represented by a random variable, X, where each number relates to the die's face value.

Random variables are classified into two main types: discrete and continuous. Discrete random variables take on a countable number of distinct values, much like the integers 1, 2, 3, and so on. In contrast, continuous random variables can take on any value within a given range or interval. The exercise focuses on discrete random variables, specifically those with integer values, which align with the type of data one would encounter when dealing with counts or whole units.

Probability Mass Function (PMF)

The probability mass function (pmf) is a function that gives us the probability that a discrete random variable is exactly equal to some value. It's a crucial concept when dealing with discrete random variables as it provides a complete description of the probability distribution. For any given value x, the pmf is denoted as p(x), signifying the probability that our random variable X takes on the value x.

In our particular exercise, the pmf given is \(p(x_1, x_2)= (1/2)^{x_1+x_2}\), which describes the probability of the random variables \(X_1\) and \(X_2\) for the integers starting from 1 to infinity. The pmf guides us in calculating other statistical qualities of the random variables, such as the moment generating function (mgf), expectations, and variances, by articulating how the probabilities are distributed across the different values the random variables can assume.

Geometric Series Convergence

When we talk about geometric series convergence, we refer to a situation where the sum of the terms of a geometric series approaches a finite limit. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the 'common ratio'.

For example, in the series \(1, r, r^2, r^3, ...\), r is the common ratio. For the series to converge, the absolute value of the common ratio must be less than one \( (|r| < 1)\). This condition ensures the terms become smaller and smaller, adding progressively less to the sum and leading the series to converge to a finite number. In our exercise, the series converges if \(|e^{t_i}/2|<1\), which implies \(|t_i|< ln(2)\), setting the bounds for our moment generating function calculations.

Mathematical Statistics

Mathematical statistics is a branch of mathematics that deals with the collection, analysis, interpretation, and presentation of data. It uses probability theory to model the randomness inherent in the data and applies statistical theories and methods to analyze and make inferences from the data.

In our exercise, the field of mathematical statistics is applied to determine the properties of the joint moment generating function (mgf) for two discrete random variables. The mgf itself is a significant tool in this field, as it provides a compact representation of all the moments (expected values of powers) of a random variable. Proving that the joint mgf of the variables can be expressed as the product of their individual mgfs, as demonstrated in the step-by-step solution, is a powerful result that allows statisticians and mathematicians to handle the random variables separately, simplifying both computations and theoretical work.