Question

Let \(X_{1}\) and \(X_{2}\) be two independent random variables. Suppose that \(X_{1}\) and \(Y=X_{1}+X_{2}\) have Poisson distributions with means \(\mu_{1}\) and \(\mu>\mu_{1}\), respectively. Find the distribution of \(X_{2}\).

Short answer

Therefore, \(X_{2}\) will have a Poisson distribution with mean \(\mu - \mu_{1}\).

Step-by-step solution

Recognize the Poisson Distribution Property

If \(X_{1}\) and \(X_{2}\) are two independent Poisson random variables with respective parameters \(\mu_{1}\) and \(\mu_{2}\), then the sum \(Y = X_{1} + X_{2}\) also follows a Poisson distribution, the parameter of which is the sum of the original parameters, that is, \(\mu = \mu_{1} + \mu_{2}\).

Subtraction of Means

In order to find the rate parameter \(\mu_{2}\) for \(X_{2}\), we simply subtract \(\mu_{1}\) from \(\mu\) as we have the relation \(\mu = \mu_{1} + \mu_{2}\). So, \(\mu_{2} = \mu - \mu_{1}\).

Find the Distribution of \(X_{2}\)

Given the rate parameter \(\mu_{2}\), we can then detail the distribution of \(X_{2}\). Since \(X_{2}\) is a Poisson random variable, it follows a Poisson distribution with mean \(\mu_{2}\). Thus, \(X_{2}\) is Poisson-distributed with parameter \(\mu_{2}\).

Key concepts

Random Variables

A random variable is a mathematical function that maps outcomes of a random process to numerical values. Essentially, it assigns numbers to the outcomes of random events such as tossing a coin or rolling a die. Random variables can be discrete or continuous, depending on the type of outcomes they represent.

Discrete random variables take on a countable number of distinct values. An example is the roll of a die, which can result in one of six possible outcomes. Continuous random variables, on the other hand, can take on any value within an interval, such as the time it takes for a radioactive particle to decay.

The Poisson random variable is a type of discrete random variable commonly used to model the number of occurrences of an event over a specified interval of time or space. The events being modeled are typically rare and independent, such as the number of phone calls a receptionist receives in an hour or the number of stars in a certain region of the sky.

Independent Poisson Random Variables

Independence is a key concept when dealing with multiple random variables. Two random variables are independent if the occurrence of one event does not affect the probability of occurrence of the other. In other words, knowing the outcome of one does not provide any information about the outcome of the other.

In the context of Poisson random variables, independence implies that the number of occurrences for one random variable has no impact on the number of occurrences for another. This property is crucial in determining the combined behavior of two separate Poisson processes.

When two independent Poisson random variables are added together, as in the given exercise, the result is also a Poisson random variable. The parameter of this new Poisson distribution is the sum of the parameters of the individual distributions. This is a unique property of Poisson distributions that greatly simplifies the process of finding resulting distributions in scenarios involving several independent Poisson processes.

Distribution Properties

The properties of a distribution describe the behavior of a random variable. Two important properties are the mean and variance. The mean, often denoted by \(\mu\), provides the 'center' of the distribution, while the variance measures the spread of values around the mean.

The Poisson distribution is characterized by its mean, which also serves as its variance, illustrating that the average number of events and the variability of those events are equal. Specifically, if a random variable \(X\) follows a Poisson distribution with parameter \(\mu\), then its mean and variance are both equal to \(\mu\).

An essential distribution property used in solving the given exercise is that the sum of independent Poisson random variables is itself a Poisson random variable with a mean equal to the sum of the individual means. Utilizing these distribution properties allows for the determination of unknown parameters and understanding the behavior of complex stochastic processes.