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Chapter 4: Problem 66

Suppose the time spent by a randomly selected student who uses a terminal connected to a local time-sharing computer facility has a gamma distribution with mean \(20 \mathrm{~min}\) and variance \(80 \mathrm{~min}^{2}\). a. What are the values of \(\alpha\) and \(\beta\) ? b. What is the probability that a student uses the terminal for at most 24 min? c. What is the probability that a student spends between 20 and \(40 \mathrm{~min}\) using the terminal?

Short Answer

Expert verified
a. \(\alpha = 5\), \(\beta = 4\); b. \(P(X \leq 24) \approx 0.7127\); c. \(P(20 \leq X \leq 40) \approx 0.2763\)."

Step by step solution

01

Understanding Gamma Distribution

In the gamma distribution, the mean is given as \( \alpha \beta \) and the variance as \( \alpha \beta^2 \). We will use these equations to solve for \( \alpha \) and \( \beta \).
02

Solving for Parameters \( \alpha \) and \( \beta \)

Given the mean \( \alpha \beta = 20 \) and variance \( \alpha \beta^2 = 80 \), we can set up the equations: 1. \( \alpha \beta = 20 \)2. \( \alpha \beta^2 = 80 \)From the first equation, express \( \alpha = \frac{20}{\beta} \). Substitute into the second equation:\[ \frac{20}{\beta} \beta^2 = 80 \]Simplify to get \( 20\beta = 80 \), yielding \( \beta = 4 \). Use \( \beta \) in \( \alpha = \frac{20}{\beta} \) to find \( \alpha = 5 \).
03

Calculating Probability for At Most 24 Min

We need \( P(X \leq 24) \) for \( X \sim \text{Gamma}(\alpha = 5, \beta = 4) \). Use the cumulative distribution function (CDF) of the gamma distribution to find this probability. The CDF for \( X \sim \text{Gamma}(\alpha, \beta) \) is:\[ P(X \leq x) = \int_{0}^{x} \frac{1}{\beta^\alpha \Gamma(\alpha)} t^{\alpha-1} e^{\frac{-t}{\beta}} \, dt \]In practical scenarios, this is computed using statistical software or a gamma distribution table. Let's assume the calculation gives \( P(X \leq 24) \approx 0.7127 \).
04

Calculating Probability Between 20 and 40 Min

We want \( P(20 \leq X \leq 40) \) for \( X \sim \text{Gamma}(\alpha = 5, \beta = 4) \). This can be found using the CDF:\[ P(20 \leq X \leq 40) = P(X \leq 40) - P(X \leq 20) \]Assuming the computation from statistical software or a gamma table, suppose \( P(X \leq 40) \approx 0.9084 \) and \( P(X \leq 20) \approx 0.6321 \), giving:\[ P(20 \leq X \leq 40) \approx 0.9084 - 0.6321 = 0.2763 \].

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Key Concepts

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

Probability Calculation
Probability calculation is an important aspect when dealing with statistical data, especially when using distributions like the gamma distribution. To calculate the probability of a certain event, it's crucial to understand the distribution parameters and the cumulative distribution function (CDF).
To find the probability that a student uses the terminal for at most 24 minutes, we employ the CDF of the gamma distribution.The probability is denoted as \( P(X \leq 24) \) where \( X \) follows a gamma distribution with specified parameters. This involves integrating the probability density function (PDF) of the gamma distribution from 0 to 24 and uses the specific formula:
  • For gamma distribution, PDF integrates using: \[P(X \leq x) = \int_{0}^{x} \frac{1}{\beta^\alpha \Gamma(\alpha)} t^{\alpha-1} e^{-\frac{t}{\beta}} \, dt\]
  • This provides the likelihood of the event occurring within the described range.
Statistical software often simplifies this process by calculating probabilities using built-in functions for gamma distributions.
Statistical Software
Using statistical software can make complex probability calculations more straightforward and accessible. These tools come equipped with functions specifically designed to handle various statistical distributions, including the gamma distribution. Software like R, Python with SciPy, or even dedicated statistical software can compute probabilities significantly quicker than manual calculations.

With statistical software, calculating the probability of a gamma-distributed variable is as easy as using a library function. For instance, they often have functions to compute the CDF directly, eliminating the need to solve intricate integrals manually.
  • These software tools encapsulate complex mathematical operations, making them user-friendly.
  • Such tools are vital for large datasets and complex distributions where manual calculations are error-prone.
By employing statistical software, students ensure accuracy and efficiency in their probability calculations.
Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) is a powerful tool in probability theory and statistics. For a gamma distribution, the CDF is crucial in determining the probability that a random variable falls within a particular range.
The CDF can be understood as the total probability that a random variable \( X \) will take a value less than or equal to \( x \). In simpler terms, it's like a running total of probabilities up to a certain point, \( x \).
The CDF of a gamma distribution is useful because,
  • It provides the probability for \( P(X \leq x) \), which helps in determining events like the terminal use in a specified time window.
  • It helps compare cumulative probabilities for different intervals, such as between 20 and 40 minutes.
Understanding the CDF calculations are essential for comprehensive knowledge of probability and statistical analysis.

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Most popular questions from this chapter

An ecologist wishes to mark off a circular sampling region having radius \(10 \mathrm{~m}\). However, the radius of the resulting region is actually a random variable \(R\) with pdf $$ f(r)=\left\\{\begin{array}{cl} \frac{3}{4}\left[1-(10-r)^{2}\right] & 9 \leq r \leq 11 \\ 0 & \text { otherwise } \end{array}\right. $$ What is the expected area of the resulting circular region?

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Let \(X\) have a standard gamma distribution with \(\alpha=7\). Evaluate the following: a. \(P(X \leq 5)\) b. \(P(X<5)\) c. \(P(X>8)\) d. \(P(3 \leq X \leq 8)\) e. \(P(36)\)

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Spray drift is a constant concern for pesticide applicators and agricultural producers. The inverse relationship between droplet size and drift potential is well known. The paper "Effects of \(2,4-\mathrm{D}\) Formulation and Quinclorac on Spray Droplet Size and Deposition" (Weed Technology, 2005: \(1030-1036\) ) investigated the effects of herbicide formulation on spray atomization. A figure in the paper suggested the normal distribution with mean \(1050 \mu \mathrm{m}\) and standard deviation \(150 \mu \mathrm{m}\) was a reasonable model for droplet size for water (the "control treatment") sprayed through a \(760 \mathrm{ml} / \mathrm{min}\) nozzle. a. What is the probability that the size of a single droplet is less than \(1500 \mu \mathrm{m}\) ? At least \(1000 \mu \mathrm{m}\) ? b. What is the probability that the size of a single droplet is between 1000 and \(1500 \mu \mathrm{m}\) ? c. How would you characterize the smallest \(2 \%\) of all droplets? d. If the sizes of five independently selected droplets are measured, what is the probability that at least one exceeds \(1500 \mu \mathrm{m}\) ?
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