The response times on an online computer terminal have...

Name: Problem 106, Chapter 4: Mathematical Statistics with Applications
Uploaded: 2021-05-03T22:25:24-08:00
Duration: 1 min 52 s
Channel: Jerrah Biggerstaff
Description: The response times on an online computer terminal have approximately a gamma distribution with mean four seconds and variance eight seconds? a. Write the probability density function for the response times. b. What is the probability that the response time on the terminal is less than five seconds?

The response times on an online computer terminal have approximately a gamma distribution with mean four seconds and variance eight seconds? a. Write the probability density function for the response times. b. What is the probability that the response time on the terminal is less than five seconds?

The response times on an online computer terminal have approximately a gamma distribution with mean four seconds and variance eight seconds?
a. Write the probability density function for the response times.
b. What is the probability that the response time on the terminal is less than five seconds?

Key Concepts

Gamma Distribution

The gamma distribution is a continuous probability distribution characterized by a shape parameter and a scale parameter. It is widely used in scenarios such as waiting times and response times, where the distribution of the time until an event occurs is modeled. Its probability density function is defined using the gamma function, which is a continuous extension of the factorial function.

Parameter Estimation for Gamma Distribution

Parameters of the gamma distribution are estimated based on the relationships between the mean and variance. Specifically, the mean is equal to the product of the shape and scale parameters, and the variance is equal to the shape parameter multiplied by the square of the scale parameter. These relationships allow one to solve for the parameters when the first two moments of the data are known.

Probability Density Function (PDF)

The probability density function is a mathematical function that describes the likelihood of a continuous random variable taking on a particular value. For the gamma distribution, the pdf includes the gamma function and the parameters derived from the mean and variance. Understanding the PDF is crucial for deriving probabilities over specified intervals.

Cumulative Distribution Function (CDF) and Tail Probability Calculation

The cumulative distribution function provides the probability that a random variable is less than or equal to a particular value. In the context of the gamma distribution, the CDF is obtained by integrating the PDF from zero up to that value. It is indispensable for calculating probabilities such as the likelihood of a response time being below a given threshold.

Transcript

Please give Ace some feedback