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Chapter 6: Problem 3

For a continuous probability distribution, why is \(P(a

Short Answer

Expert verified
In a continuous probability distribution, the probability that \(x\) falls in an open interval (a, b) is the same as the probability that \(x\) falls in the closed interval [a, b] because the probability of \(x\) taking on any specific value a or b (the endpoints) is 0. Thus, the inclusion/exclusion of the endpoints does not impact the overall probability.

Step by step solution

01

Understand continuous probability distribution

In a continuous probability distribution, the probability that the random variable \(X\) takes on any specific value x is 0. This is because there are an infinite number of possible values x can take on. Because of this, we typically talk about the probability that \(X\) falls within a certain range or interval of values, rather than taking on a specific value.
02

Definition of Probability for continuous distributions

The probability that \(X\) lies in an interval (a, b) is the integral from a to b of the probability density function \(f(x)\) of \(X\). Similarly, the probability that \(X\) lies in the interval [a, b] is also the integral from a to b of \(f(x)\). Here, (a, b) is an open interval where endpoints a and b are not included, while [a, b] is a closed interval where endpoints a and b are included.
03

Equivalence of probabilities

The probability of the random variable \(X\) falling within the interval [a, b] is equal to the probability of \(X\) falling within the interval (a, b) because the probability of \(X\) taking on any specific value is 0. Thus, adding the endpoints a and b to the open interval (a, b) does not change the probability.

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Key Concepts

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

Probability Density Function
A probability density function (PDF) is crucial in describing the behavior of a continuous random variable. It's like a curve that shows how the values of the random variable are distributed. Imagine the PDF as a smooth, continuous line over a plot, where each point on the line tells you how likely it is for the random variable to be near that point.

The area under the PDF curve within a specified interval gives you the probability that the variable falls within that interval. This means if you're looking at the PDF, you're really looking at probabilities. Remember:
  • A PDF is not a probability at a specific point but a way to find probabilities over ranges.
  • The entire area under a PDF curve equals 1, meaning it accounts for all possible outcomes.
Probability Intervals
Probability intervals are about finding the probability that a continuous random variable lies between two values. These intervals can be open, like (a, b), which means the endpoints are not included, or closed, like [a, b], where the endpoints are included.

For continuous distributions, the intervals (a, b) and [a, b] yield the same probability. This is because, in continuous distributions, the precise probability of landing exactly on a point, like 'a' or 'b,' is always zero. So, whether you include or exclude these endpoints doesn't change the probability for practical purposes.
  • Open interval (a, b): Points 'a' and 'b' are not counted.
  • Closed interval [a, b]: Points 'a' and 'b' are counted, but it doesn鈥檛 affect the probability result.
Continuous Random Variable
A continuous random variable is one that can take any value within a given range. These variables are characterized by having an infinite number of possible outcomes within a range.

Imagine measuring the exact height of a person. This height could be any number within a range, like between 150 cm to 200 cm, and could even include decimals. It's like looking at smooth, seamless data. Common examples include:
  • Height
  • Temperature
  • Time
These aspects make continuous random variables less about exact numbers and more about intervals or ranges.
Integration in Probability
Integration is a mathematical process that comes in very handy when working with continuous probability distributions. It's the tool that allows us to calculate probabilities from a probability density function.

Think of integration as a way to sum up small probabilities along a stretch of values. When you integrate a PDF over an interval from 'a' to 'b,' you're effectively adding up all the tiny probabilities of your continuous random variable falling within that range.
  • It sums probabilities over an interval, not at specific points.
  • The integral of the entire probability density function over all spaces results in 1, representing the certainty of all possible outcomes happening.

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Most popular questions from this chapter

Hurbert Corporation makes font cartridges for laser printers that it sells to Alpha Electronics Inc. The cartridges are shipped to Alpha Electronics in large volumes. The quality control department at Alpha Electronics randomly selects 100 cartridges from each shipment and inspects them for being good or defective. If this sample contains 7 or more defective cartridges, the entire shipment is rejected. Hurbert Corporation promises that of all the cartridges, only \(5 \%\) are defective. a. Find the probability that a given shipment of cartridges received by Alpha Electronics will be accepted. b. Find the probability that a given shipment of cartridges received by Alpha Electronics will not be accepted.

Determine the following probabilities for the standard normal distribution. a. \(P(-2.46 \leq z \leq 1.88)\) b. \(P(0 \leq z \leq 1.96)\) c. \(P(-2.58 \leq z \leq 0)\) d. \(P(z \geq .73)\)

Fast Auto Service provides oil and lube service for cars. It is known that the mean time taken for oil and lube service at this garage is 15 minutes per car and the standard deviation is \(2.4\) minutes. The management wants to promote the business by guaranteeing a maximum waiting time for its customers. If a customer's car is not serviced within that period, the customer will receive a \(50 \%\) discount on the charges. The company wants to limit this discount to at most \(5 \%\) of the customers. What should the maximum guaranteed waiting time be? Assume that the times taken for oil and lube service for all cars have a normal distribution.

A psychologist has devised a stress test for dental patients sitting in the waiting rooms. According to this test, the stress scores (on a scale of 1 to 10 ) for patients waiting for root canal treatments are found to be approximately normally distributed with a mean of \(7.59\) and a standard deviation of \(.73\). a. What percentage of such patients have a stress score lower than \(6.0\) ? b. What is the probability that a randomly selected root canal patient sitting in the waiting room has a stress score between \(7.0\) and \(8.0\) ? c. The psychologist suggests that any patient with a stress score of \(9.0\) or higher should be given a sedative prior to treatment. What percentage of patients waiting for root canal treatments would need a sedative if this suggestion is accepted?

Find the following binomial probabilities using the normal approximation. a. \(n=140, \quad p=.45, \quad P(x=67)\) b. \(n=100, \quad p=.55, \quad P(52 \leq x \leq 60)\) c. \(n=90, \quad p=.42, \quad P(x \geq 40)\) d. \(n=104, p=.75, \quad P(x \leq 72)\)
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