91Ó°ÊÓ

We have seen that if \(E\left(X_{1}\right)=E\left(X_{2}\right)=\ldots=E\left(X_{n}\right)=\mu\), then \(E\left(X_{1}+\cdots+X_{n}\right)=n \mu\). In some applications, the number of \(X_{i}\) 's under consideration is not a fixed number \(n\) but instead is an rv \(N\). For example, let \(N=\) the number of components that are brought into a repair shop on a particular day, and let \(X_{i}\) denote the repair shop time for the \(i\) th component. Then the total repair time is \(X_{1}+X_{2}+\cdots+X_{N}\), the sum of a random number of random variables. When \(N\) is independent of the \(X_{i}\) 's, it can be shown that $$ E\left(X_{1}+\cdots+X_{N}\right)=E(N) \cdot \mu $$ a. If the expected number of components brought in on a particularly day is 10 and expected repair time for a randomly submitted component is \(40 \mathrm{~min}\), what is the expected total repair time for components submitted on any particular day? b. Suppose components of a certain type come in for repair according to a Poisson process with a rate of 5 per hour. The expected number of defects per component is 3.5. What is the expected value of the total number of defects on components submitted for repair during a 4-hour period? Be sure to indicate how your answer follows from the general result just given.

Step by step solution

01

Identify Given Values for Part (a)

For part (a), we are given that the expected number of components, \( E(N) \), is 10, and the expected repair time per component, \( \mu \), is 40 minutes.

02

Apply General Expectation Formula for Part (a)

The formula to find the expected total repair time is \( E(X_1 + X_2 + \cdots + X_N) = E(N) \cdot \mu \). Substituting the values, we get \( E(N) = 10 \) and \( \mu = 40 \).

03

Calculate Expected Total Repair Time for Part (a)

Compute \( 10 \cdot 40 = 400 \) minutes. Therefore, the expected total repair time on any particular day is 400 minutes.

04

Define Values for Poisson Process in Part (b)

For part (b), the components arrive according to a Poisson process with a rate (\( \lambda \)) of 5 per hour over 4 hours. Therefore, the expected number of components brought in is \( E(N) = 5 \times 4 = 20 \).

05

Find Expected Defects for Part (b)

Each component is expected to have 3.5 defects on average, making \( \mu = 3.5 \).

06

Apply General Expectation Formula for Part (b)

The formula \( E(X_1 + X_2 + \cdots + X_N) = E(N) \cdot \mu \) is used again. Substituting \( E(N) = 20 \) and \( \mu = 3.5 \), we calculate the total expected defects: \( 20 \cdot 3.5 = 70 \).

07

Conclude Expected Total Number of Defects for Part (b)

The expected total number of defects on components submitted for repair during a 4-hour period is 70.

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Key Concepts

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

Random Variables

A random variable is a fundamental concept in probability theory. It is a variable whose possible values result from a random phenomenon. Random variables can be either discrete or continuous. A discrete random variable has specific, separate values, like the number of cars that pass through a toll booth in an hour. A continuous random variable, on the other hand, can have any value within an interval, like the time it takes for a student to finish a test.
In problems involving random variables, like the exercise given, we often deal with sums of random variables. Understanding how to manipulate these sums and their properties is key to solving many probability and statistics problems. The exercise shows how expected values for a sum of random variables can be computed when the number of random variables, such as these repair times, is itself a random variable.

Poisson Process

A Poisson process is a statistical process that models events occurring randomly over a fixed period of time or space. It is characterized by its rate parameter, often denoted as \( \lambda \), which indicates the average number of events occurring in a given time period. In the context of the exercise, we're dealing with repairs coming into a shop, which fits a Poisson process.
The Poisson process is commonly used in scenarios where we want to predict the number of events happening in a given timeframe, such as phone calls received by a call center, customers entering a store, or defects found in production. In practical application, as shown in the exercise, the Poisson process allowed us to calculate the expected number of repair components arriving at the shop over a four-hour period by using the formula: expected value = \( \lambda \times \text{time} \).

Expectation Formula

The expectation formula, or expected value, provides a measure of the central tendency of a random variable. In simpler terms, it's the long-term average value that a random variable assumes through its possible outcomes. The formula \( E(X) = \sum x_i P(x_i) \) is used for discrete variables, where the sum is taken over all possible outcomes \( x_i \) multiplied by their probability \( P(x_i) \).
In our exercise, the expectation formula is adapted to accommodate a random number of items, \( N \), such that the expected value of the sum is the expected number \( E(N) \) times the average value \( \mu \). This approach greatly simplifies problems where the overall quantity is influenced by a stochastic, or randomly determined, number of instances.

Probability Theory

Probability theory is the branch of mathematics that deals with the analysis and determination of probabilities associated with different random phenomena. It provides the mathematical foundation for much of statistics and data analysis, including crucial concepts like random variables and expected values.
Utilizing probability theory, we can model uncertainty and variability in various domains, from casinos to weather forecasting. In our specific example, probability theory enables us to determine the expected total time of repair services or defects per component. It offers the tools to combine multiple random events into a coherent mathematical framework, predicting outcomes and making data-informed decisions.
Key principles underpinning these probability assessments define how we handle dependent and independent events, thus aiding our understanding of joint probabilities and conditional probabilities. Through these concepts, we gain insights into how we can expect systems to behave under uncertainty.