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Chapter 4: Problem 16

Let \(X\) be a Poisson random variable with parameter \(\lambda\). Show that \(P\\{X=i\\}\) increases monotonically and then decreases monotonically as \(i\) increases, reaching its maximum when \(i\) is the largest integer not exceeding \(\lambda\).

Short Answer

Expert verified
In summary, for a Poisson random variable X with parameter 位, the PMF P{X=i} increases monotonically for i < 位 - 1, reaches its maximum at i = 位 - 1 (the largest integer not exceeding 位), and then decreases monotonically for i > 位 - 1. This is based on analyzing the ratio \(\frac{\lambda}{i+1}\), which gives the change in consecutive PMF values.

Step by step solution

01

Recall the Poisson PMF

Recall that the probability mass function of a Poisson random variable X with parameter 位 is given by: \(P\{X=i\} = \frac{e^{-\lambda}\lambda^i}{i!}\), for i = 0, 1, 2, .... This is the function we are analyzing.
02

Calculate the ratio of consecutive PMF values

Calculate the ratio \(\frac{P\{X = i+1\}}{P\{X = i\}}\), which gives us information about whether the PMF is increasing or decreasing. \(\frac{P\{X = i+1\}}{P\{X = i\}} = \frac{\frac{e^{-\lambda}\lambda^{i+1}}{(i+1)!}}{\frac{e^{-\lambda}\lambda^i}{i!}}\) Simplify the above expression: \(\frac{P\{X = i+1\}}{P\{X = i\}} = \frac{\lambda^{i+1}i!}{\lambda^i (i+1)!}\) Cancel the common terms: \(\frac{P\{X = i+1\}}{P\{X = i\}} = \frac{\lambda}{i+1}\)
03

Analyze the Ratio

Analyze the behavior of the ratio for different values of i: 1. If i < 位 - 1, then \(\frac{\lambda}{i+1} > 1\), which implies that the PMF is increasing for i < 位 - 1. 2. If i = 位 - 1, then \(\frac{\lambda}{i+1} = 1\) and the PMF is at its maximum. 3. If i > 位 - 1, then \(\frac{\lambda}{i+1} < 1\), which implies that the PMF is decreasing for i > 位 - 1. Based on the ratio \(\frac{\lambda}{i+1}\), the PMF P{X=i} increases monotonically as i increases up to i = 位 - 1 and then decreases monotonically as i increases further. The maximum value of the PMF occurs when i is the largest integer not exceeding 位.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Mass Function
Understanding the probability mass function (PMF) is crucial when working with Poisson distributions. The PMF provides the probabilities of different outcomes or events for a Poisson-distributed random variable.
For a Poisson random variable \(X\) with parameter \(\lambda\), the PMF is given by:
  • \(P\{X=i\} = \frac{e^{-\lambda} \lambda^i}{i!}\)
This equation tells us the likelihood that the random variable \(X\) takes on the value \(i\).
Key elements in the PMF include \(e^{-\lambda}\), which represents the exponential decay, \(\lambda^i\), which scales the probability based on the parameter \(\lambda\), and the factorial \(i!\), which adjusts for the number of occurrences.
Monotonicity
Monotonicity in probabilities refers to how the probability values change as the variable \(i\) changes. For the Poisson distribution, the PMF shows a special pattern of increasing and then decreasing values.
To determine this pattern, we assess the ratio of consecutive probability masses, say for \(i\) and \(i + 1\):
  • \(\frac{P\{X = i+1\}}{P\{X = i\}} = \frac{\lambda}{i+1}\)
Analyzing this ratio helps us see when the PMF is increasing or decreasing:
  • If \(i < \lambda - 1\), the ratio is greater than 1, meaning the PMF increases.
  • If \(i = \lambda - 1\), the ratio equals 1, and the PMF reaches its peak.
  • If \(i > \lambda - 1\), the ratio is less than 1, indicating the PMF decreases.
This pattern illustrates that the PMF is initially increasing and then decreasing with changes in \(i\).
Maximum Likelihood
Maximum likelihood in this context refers to finding the value of \(i\) which maximizes the probability \(P\{X = i\}\) for a Poisson distribution.
From our analysis, we found that the probability reaches its maximum when \(i\) is close to \(\lambda\), specifically when \(i\) is the largest integer not exceeding \(\lambda\).
Finding this point ensures that the probability of observing exactly \(i\) occurrences is at its highest, given the parameter \(\lambda\). This understanding is crucial for making inferences or predictions based on observed data.
Parameter Analysis
When dealing with Poisson distributions, parameter analysis focuses on how the parameter \(\lambda\) affects the probabilities and behavior of the distribution.
\(\lambda\) is the rate at which events occur in a fixed interval and directly influences:
  • The location of the PMF's maximum, as it is typically near \(\lambda\).
  • The shape of the PMF, influencing how quickly probabilities increase or decrease.
For instance, a higher \(\lambda\) leads to the probabilities spreading out over a larger range of \(i\), often causing the peak to shift rightwards.
Understanding how \(\lambda\) changes the distribution's character can help in appropriately modeling real-world phenomena where event rates are known or can be estimated.

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Most popular questions from this chapter

An urn initially contains one red and one blue ball. At each stage a ball is randomly chosen and then replaced along with another of the same color. Let \(X\) denote the selection number of the first chosen ball that is blue. For instance, if the first selection is red and the second blue, then \(X\) is equal to 2 . (a) Find \(P\\{X>i\\}, i \geq 1\). (b) Show that with probability 1 , a blue ball is eventually chosen. (That is, show that \(P\\{X<\infty\\}=1 .\) ) (c) Find \(E[X]\).

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