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Chapter 5: Problem 9

A continuous probability function is restricted to the portion between \(x=0\) and \(7 .\) What is \(P(x=10) ?\)

Short Answer

Expert verified
The probability \( P(x=10) = 0 \) because it is outside the specified range \( 0 \leq x \leq 7 \).

Step by step solution

01

Understanding the Problem

We are given a continuous probability function restricted to the range between \( x=0 \) and \( x=7 \). The problem asks for the probability \( P(x=10) \), which is outside the given range.
02

Identifying Properties of Continuous Probability Functions

A continuous probability function, like a probability density function (PDF), assigns probabilities over an interval, not specific points. Therefore, for a single point the probability is actually zero.
03

Conclusion about Probability Outside the Restricted Range

Since the function is restricted to the interval \( x \in [0, 7] \), any \( x \) value outside this range (like \( x=10 \)) has zero probability. Hence, \( P(x=10) = 0 \).

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Key Concepts

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

Understanding Probability Density Functions
A Probability Density Function (PDF) is central to the concept of continuous probability distributions. Unlike discrete distributions, which assign probabilities to individual outcomes, a PDF describes the probability of a continuous random variable falling within a certain range of values. This is done by defining a function, often denoted as \( f(x) \), over a certain interval.
  • The PDF itself does not give the probability at a single point. Instead, it is used to calculate the probability over a range.
  • For a given continuous random variable \( X \), the probability that \( X \) lies within the interval \([a, b]\) is given by the integral of the PDF over \([a, b]\), written as \( P(a \leq X \leq b) = \int_a^b f(x) \, dx \).
  • The total area under the PDF curve across the entire possible range of values is equal to 1, signifying the certainty that the random variable takes on a value within this range.
Exploring Range Restrictions in Continuous Probability
When dealing with continuous random variables, it is crucial to identify the range over which these variables are defined. Range restrictions define the specific interval where the probability density function is applicable.
  • Any value of \( x \) that lies outside the defined range has a probability of zero. This is because the PDF is not defined outside this interval.
  • For example, if a PDF is restricted to the range \([0, 7]\), calculations and interpretations rely entirely on this specified interval.
  • If we consider a continuous probability function that is restricted between \( x = 0 \) and \( x = 7 \), any point outside this range immediately renders the probability to zero, such as \( P(x=10) = 0 \).
Range restrictions are essential in probability as they determine the domain over which the PDF is mapped and ensure that calculations are meaningful within the correct context.
The Reality of Single-Point Probability in Continuous Distributions
In the realm of continuous probability distributions, the concept of single-point probability can be quite intriguing. Essentially, for any continuous random variable, the probability of observing exactly one specific value is zero.
  • This occurs because probability is only meaningful over an interval given its definition as an integral over a range.
  • Mathematically, this means that for any point \( x \), the probability \( P(X = x) \) is zero due to the properties of integration.
  • Contrast this with discrete probability, where specific points can indeed have non-zero probabilities. In continuous terms, we always refer to the probability of a range or interval.

Understanding single-point probability helps emphasize why questions such as \( P(x = 10) \) yield zero probability in continuous contexts, particularly when considering range restrictions.

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Most popular questions from this chapter

Use the following information to answer the next seven exercises. A distribution is given as \(X \sim \operatorname{Exp}(0.75).\) Draw the distribution.

Use the following information to answer the next ten questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. $$\begin{array}{|c|c|c|c|c|c|c|}\hline 1.5 & {2.4} & {3.6} & {2.6} & {1.6} & {2.4} & {2.0} \\ \hline 3.5 & {2.5} & {1.8} & {2.4} & {2.5} & {3.5} & {4.0} \\\ \hline 2.6 & {1.6} & {2.2} & {1.8} & {3.8} & {2.5} & {1.5} \\\\\hline {2.8}&{1.8} &{4.5}&{1.9} &{1.9}& {3.1}& {1.6}\\\\\hline\end{array}$$ $$\text{Table 5.4}$$ The sample mean \(=2.50\) and the sample standard deviation \(=0.8302.\) The distribution can be written as \(X \sim U(1.5,4.5).\) What type of distribution is this?

Use the following information to answer the next ten questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes. $$\begin{array}{|c|c|c|c|c|c|c|}\hline 1.5 & {2.4} & {3.6} & {2.6} & {1.6} & {2.4} & {2.0} \\ \hline 3.5 & {2.5} & {1.8} & {2.4} & {2.5} & {3.5} & {4.0} \\\ \hline 2.6 & {1.6} & {2.2} & {1.8} & {3.8} & {2.5} & {1.5} \\\\\hline {2.8}&{1.8} &{4.5}&{1.9} &{1.9}& {3.1}& {1.6}\\\\\hline\end{array}$$ $$\text{Table 5.4}$$ The sample mean \(=2.50\) and the sample standard deviation \(=0.8302.\) The distribution can be written as \(X \sim U(1.5,4.5).\) What is the height of \(f(x)\) for the continuous probability distribution?

At an urgent care facility, patients arrive at an average rate of one patient every seven minutes. Assume that the duration between arrivals is exponentially distributed. a. Find the probability that the time between two successive visits to the urgent care facility is less than 2 minutes. b. Find the probability that the time between two successive visits to the urgent care facility is more than 15 minutes. c. If 10 minutes have passed since the last arrival, what is the probability that the next person will arrive within the next five minutes? d. Find the probability that more than eight patients arrive during a half- hour period.

For a continuous probablity distribution, \(0 \leq x \leq 15 .\) What is \(P(x > 15) ?\)
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