Problem 107 - Applied Statistics and Probability for Engineers

Question

An article in Atmospheric Chemistry and Physics ["Relationship Between Particulate Matter and Childhood Asthma - Basis of a Future Warning System for Central Phoenix," 2012 , Vol. \(12,\) pp. \(2479-2490]\) linked air quality to childhood asthma incidents. The study region in central Phoenix, Arizona recorded 10,500 asthma incidents in children in a 21 -month period. Assume that the number of asthma incidents follows a Poisson distribution. (a) Approximate the probability of more than 550 asthma incidents in a month. (b) Approximate the probability of 450 to 550 asthma incidents in a month. (c) Approximate the number of asthma incidents exceeded with probability \(5 \%\). (d) If the number of asthma incidents was greater during the winter than the summer, what would this imply about the Poisson distribution assumption?

Step-by-Step Solution

Answer

(a) Use Poisson CDF to find \( P(X > 550) \). (b) Find \( P(450 \leq X \leq 550) \). (c) Determine \( k \) for the 95th percentile. (d) Seasonal variation suggests non-Poisson variation.

Step 1: Determine Monthly Average Rate (λ)

First, calculate the average number of asthma incidents per month. With a total of 10,500 incidents over 21 months, divide 10,500 by 21. Thus, the rate \( \lambda \) for a month is:\[ \lambda = \frac{10500}{21} = 500 \] incidents per month.

Step 2: Calculate Probability for More than 550 Incidents using Poisson Approximation

We want the probability of more than 550 incidents in a month. In Poisson distribution terms, this is:\[ P(X > 550) = 1 - P(X \leq 550) \]Using the Poisson cumulative distribution function (CDF), calculate \( P(X \leq 550) \), then subtract it from 1 to find \( P(X > 550) \).\This requires a calculator or software that provides Poisson distribution calculations.

Step 3: Calculate Probability Range for 450 to 550 Incidents

To find the probability of between 450 to 550 incidents:\[ P(450 \leq X \leq 550) = P(X \leq 550) - P(X < 450) \]Use the Poisson CDF to find \( P(X \leq 550) \) and \( P(X < 450) \) (which is \( P(X \leq 449) \)), and then subtract the latter from the former to get the probability for the range.

Step 4: Determine 95th Percentile for Asthma Incidents

The 95th percentile corresponds to the number of incidents exceeded with 5% probability. In Poisson terms, find \( k \) such that \( P(X \leq k) = 0.95 \).Use a Poisson distribution table or software to determine this percentile.

Step 5: Interpret Seasonal Variability and Poisson Assumption

If incidents are greater during winter than summer, it suggests variability that Poisson distribution might not account for, as it assumes a constant average rate (\( \lambda \)). Such periodic variation could mean the data might better be modeled with a distribution that accounts for temporal trends.

Key Concepts

Understanding Probability in Poisson Distribution

Probability is a way of measuring how likely an event is to happen. In the context of a Poisson distribution, it helps to determine the likelihood of a certain number of events — like asthma incidents — occurring within a fixed interval of time.
When dealing with Poisson’s distribution for asthma incidents in this case, we calculate the probability that an event, such as more than 550 asthma cases in a month, happens. This calculation involves the Poisson cumulative distribution function (CDF), which provides the combined probability of obtaining a certain number or fewer incidents.
For instance, if we say "more than 550 incidents," we're focusing on the probability of getting 551 and above. To calculate this probability, one would determine the probability of having 550 or fewer incidents and then subtract that from 1 (the total probability). This approach capitalizes on the cumulative nature of the function.

Examining Asthma Incidents

Asthma incidents refer to episodes where individuals experience asthma attacks. In our study related to central Phoenix, we consider these incidents happening to children. The Poisson distribution is helpful because it estimates the number of episodes expected to occur over a period given a known average rate.
With a reported total of 10,500 incidents over 21 months, the study averaged about 500 incidents monthly. This average rate becomes our Lambda (\( \lambda \)), which is a crucial parameter in Poisson distribution. It signifies the expected number of events in a given period.
The steady flow of 500 incidents monthly allows us to use Poisson distribution to predict probabilities for ranges like 450 to 550 asthma incidents, offering insightful predictions on the likely range of asthma occurrences any given month.

Seasonal Variability in Asthma Cases

Seasonal variability refers to fluctuations in event frequency due to changes aligned with the seasons. It becomes interesting when discussing asthma incidents as it considers how incidents might increase in colder months.
If more asthma cases occur in winter, this indicates a deviation from the stable average rate, which the basic Poisson model assumes. The assumption of a consistent Lambda (\( \lambda \)) underlying Poisson requires constant incident rates across all times. However, real-world conditions, like those during winter, might lead to a higher number of incidents. This seasonal change suggests that our model might need adjustment or different distribution to capture the variability accurately.

• This pattern in variation could be due to actual environmental changes in season like pollution or air quality adjustments.
• Identifying these trends helps in predicting future asthma-related health surges.

Using Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) is an important tool in the realm of probability and statistics, especially when using distributions like Poisson. In this exercise, the CDF provides the probability that the number of asthma incidents does not exceed a specific value.
Mathematically, it is expressed as \( P(X \leq k) \), where \( X \) represents the number of incidents, and \( k \) is a particular value or threshold.
By knowing the CDF, one can calculate the probability of having a range of values, like when finding the probability of between 450 to 550 incidents.

• The CDF helps consolidate information by adding probabilities up to a given point, facilitating the understanding of probabilities over an interval.
• It is especially applicable when looking to find percentiles or thresholds, for example, determining the number exceeded with a 5% probability in a month.

Thus, the use of a CDF in Poisson distribution problems helps clarify various probability queries regarding frequency and likelihood of occurrences over a period.