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Chapter 6: Problem 53

Computer response time is an important application of the gamma and exponential distributions. Suppose that a study of a certain computer system reveals that the rasponse time, in seconds, has an exponential distribution with a mean of 3 seconds. (a) What is the probability that response time exceeds 5 seconds? (b) What is the probability that response time exceeds 10 seconds?

Short Answer

Expert verified
The probability that response time exceeds 5 seconds is calculated as \(1 - F(5) \approx 0.188\), and the probability that response time exceeds 10 seconds is calculated as \(1 - F(10) \approx 0.0355\).

Step by step solution

01

Calculate the Rate Parameter - \(\lambda\)

The rate parameter \(\lambda\) is calculated as the reciprocal of the mean. In this case the mean given is 3 seconds, so \(\lambda = 1 / 3 \approx 0.333\).
02

Calculate Probability that Response Time Exceeds 5 seconds

You can calculate this with the cumulative distribution function of the exponential distribution \(F(x)\) using \(x = 5\). The formula is \(F(x) = 1 - e^{-\lambda * x}\). This gives \(F(5) = 1 - e^{-0.333 * 5} = 1 - e^{-1.67}\). The result is the probability that response time is less than 5. To get the probability that response time exceeds 5 seconds, calculate the complementary probability: \(1 - F(5)\).
03

Calculate Probability that Response Time Exceeds 10 seconds

Similiary, calculate this with the cumulative distribution function of the exponential distribution \(F(x)\) using \(x = 10\). The formula \(F(x) = 1 - e^{-\lambda * x}\) gives \(F(10) = 1 - e^{-0.333 * 10} = 1 - e^{-3.33}\). Calculating the complementary probability \(1 - F(10)\) will give the probability that response time exceeds 10 seconds.

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Key Concepts

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

Gamma Distribution
The gamma distribution is a two-parameter family of continuous probability distributions. This distribution is important in various fields because it generalizes the exponential distribution and is used to model the time or space that elapses before a given number of events occur. The two parameters of the gamma distribution are the shape parameter, traditionally denoted as \(k\) or \(n\), and the scale parameter, denoted as \(\theta\).
  • Relation to Exponential Distribution: When the shape parameter \(k = 1\), the gamma distribution simplifies to the exponential distribution. This specific case is significant in scenarios involving a single event occurrence.
  • Applications: The gamma distribution is often used in queuing models, reliability testing, and understanding system behavior over time.
  • Properties: The mean of a gamma distribution is \(k\theta\), and the variance is \(k\theta^2\).
Probability Computation
Probability computation involves evaluating the likelihood of a given outcome or set of outcomes. In the context of exponential distribution, calculating probabilities often involves the cumulative distribution function (CDF).
  • CDF Formula: For an exponential distribution, the CDF is given by \(F(x) = 1 - e^{-\lambda x}\). This formula helps calculate the probability of the random variable being less than or equal to a particular value \(x\).
  • Complementary Probability: To find the probability that a random event exceeds a certain value, you calculate \(1 - F(x)\). This step is critical when determining how often a system might fail or how long a particular process may take beyond a specific time frame.
  • Applications: Knowing how to compute these probabilities is crucial in risk assessment, decision making, and planning in various industries.
Cumulative Distribution Function
The cumulative distribution function (CDF) of a random variable describes the probability that the variable will take a value less than or equal to \(x\). The CDF is a vital tool in probability and statistics as it provides a full picture of the distribution.
  • Exponential Distribution CDF: For exponential distributions, the CDF is \(F(x) = 1 - e^{-\lambda x}\). This formula is particularly useful when dealing with time-to-event data, such as survival analysis in biostatistics.
  • Usefulness: The CDF allows us to compute probability directly by substituting \(x\) values and solving. This makes it easier to answer questions about the likelihood of a variable falling within a certain range.
  • Continuous Nature: Unlike discrete distributions, the continuous nature of the CDF in exponential distributions allows for precise probability calculations over any interval.
Rate Parameter
The rate parameter, commonly denoted as \(\lambda\), is a crucial element in the exponential distribution. The value of \(\lambda\) defines the distribution's form and influences various calculations, including probability and expectation.
  • Calculation: The rate parameter is calculated as the inverse of the mean, so \(\lambda = 1/\text{mean}\). This way, it demonstrates how frequently events happen over a given time or area.
  • Significance: A higher \(\lambda\) value indicates more frequent occurrences of events, which translates to a steeper decay of the exponential function. Conversely, a smaller \(\lambda\) denotes less frequent events.
  • Applications: Assessing the rate parameter allows businesses and scientists to predict how soon an event may happen or help model various scenarios like queuing systems and reliability testing.

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Most popular questions from this chapter

A soft-drink machine is regulated so that it discharges an average of 200 milliliters per cup. If the amount of drink is normally distributed with a standard deviation equal to 15 milliliters, (a) what fraction of the cups will contain more than 224 milliliters? (b) what is the probability that a cup contains between 191 and 209 milliliters? (c) how many cups will probably overflow if 230 milliliter cups are used for the next 1000 drinks? (d) below what value do we get the smallest \(25 \%\) of the drinks?

The average rate of water usage (thousands of gallons per hour) by a certain community is known to involve the lognormal distribution with parameters \(\mu=5\) and \(a=2 .\) It is important for planning purposes to get a sense of periods of high usage. What is the probability that, for any given hour, 50,000 gallons of water are used?

The weights of a large number of miniature poodles are approximately normally distributed with a mean of 8 kilograms and a standard deviation of 0.9 kilogram. If measurements arc recorded to the nearest tenth of a kilogram, find the fraction of these poodles with weights (a) over 9.5 kilograms: (b) at most 8.6 kilograms; (c) between 7.3 and 9.1 kilograms inclusive.

Suppose that the time, in hours, taken to repair a heat pump is a random variable \(X\) having a gamma distribution with parameters \(\alpha=2\) and \(3=1 / 2\). What is the probability that the next service call will require (a) at most 1 hour to repair the heat pump? (b) at least 2 hours to repair the heat pump?

If \(20 \%\) of the residents in a U.S. city prefer a white telephone over any other color available, what is the probability that among the next 1000 telephones installed in that city (a) between 170 and 185 inclusive will be white? (b) at least 210 but not more than 225 will be white?
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