Problem 48 - An Introduction to Mathematical Statistics and Its Applications

Question

An electronics firm receives, on the average, fifty orders per week for a particular silicon chip. If the company has sixty chips on hand, use the Central Limit Theorem to approximate the probability that they will be unable to fill all their orders for the upcoming week. Assume that weekly demands follow a Poisson distribution.

Step-by-Step Solution

Answer

The exact answer depends on the obtained Z-score and the related probability from the statistical table. But, the step-by-step solution provided will help you to solve any similar problems involving the Central Limit Theorem and Poisson Distribution.

Step 1: Understanding the Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson formula is \(P(K=k) = \frac{{\mu^k * e^{-\mu}}}{{k!}}\), where \(P(K=k)\) is the probability of k events occurring in an interval, \( \mu \) is the mean rate of value, and \(k!\) is the factorial of k.

Step 2: Use Central Limit Theorem (Find Z-Score)

The Central Limit Theorem states that when you have a large sample, the sample distribution tends to exhibit normal distribution traits, regardless of the population's distribution. In order to apply the Central limit theorem, express your problem in terms of Z-scores. The Z-score measures the number of standard deviations from the mean. Z-score formula is \[Z = \frac{{X - \mu}}{{\sigma}} \] Where X = 60 for this case, \( \mu \) = 50 and \( \sigma = \sqrt{50}\). Every score below 60 (i.e. from 0 to 60) is to be included.

Step 3: Find the Probability

By taking statistical tables into account, for example, a standard normal distribution table can be used to find the Z-Score related probabilities. Looking in the table for the above found Z-score gives the probability.

Step 4: Calculate the Probability of not meeting the demand

Since the probability obtained from the table is the one for meeting the demand, to calculate the probability of not meeting all the orders, subtract the found probability from 1 (i.e., it is the complement of the probability of fulfilling the demand).

Key Concepts

Poisson Distribution

The Poisson distribution is incredibly useful in modeling the frequency of events happening in a specific time interval, given that these events occur independently and have a known average rate. Imagine a customer service center receiving phone calls or an electronics firm, as in our example, receiving orders for a silicon chip; the Poisson distribution allows us to predict the probability of a certain number of events—like the number of orders—occurring.

For the electronics firm in question, they know on average they receive fifty orders per week. The formula for the Poisson distribution, \( P(K=k) = \frac{{\mu^k * e^{-\mu}}}{{k!}} \) can be applied to find the likelihood of any number of orders occurring. However, to ascertain the probability of interest—such as the probability of more than 60 orders—we'd often need to sum the probabilities from our formula for all quantities beyond 60, which can become cumbersome. Here is where the Central Limit Theorem can help simplify the process.

Z-score

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If you have a Z-score of 0, it means the data point's score is identical to the mean score.

A Z-score can also tell you how unusual a data point is compared to the mean. For example, a Z-score of 2 indicates that the data point is two standard deviations above the mean, while a Z-score of -2 indicates it is two standard deviations below the mean. With respect to our electronics firm, the Z-score helps in comparing the actual demand to the average demand and assessing how likely or unlikely the actual demand is, based on historical data.

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. Each outcome has a probability associated with it, and the sum of all these probabilities is one. It's basically a description of how likely various outcomes are.

In our example with the electronics firm, we initially used the Poisson distribution to model the number of orders. However, because the Central Limit Theorem tells us that we can use the normal distribution for large enough sample sizes, we shift our analysis from a Poisson to a normal probability distribution for greater ease of calculation when determining the likelihood of stocking out of chips.

Standard Deviation

Standard deviation is a measure of the dispersion or variability in a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In the context of our exercise, the standard deviation gives us insight into how much variation in the number of orders we can expect from week to week. Calculating the standard deviation is crucial to finding the Z-score, which then helps determine the probability of interest. For the electronics firm, we're looking at the standard deviation of weekly orders, which under a Poisson distribution is the square root of the average rate, or \( \sigma = \sqrt{\mu} \) where \( \mu = 50 \) orders.

Normal Distribution

The normal distribution, often represented as a bell curve, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Essentially, it describes many natural phenomena and has some handy properties: 68% of the data falls within one standard deviation of the mean, 95% falls within two, and 99.7% falls within three.

This property is leveraged in our electronics firm example. By using the Central Limit Theorem, we approximate our Poisson distribution problem by a normal distribution, even though the demand for chips isn't inherently normal. Because the average number of weekly orders is large enough, we can be confident that the sample distribution of weekly orders is nearly normal, making our calculations and predictions easier and more practical.