Chapter 4: Problem 11
Find the probability that the range of a random sample of size 4 from the
uniform distribution having the pdf \(f(x)=1,0
Short Answer
Expert verified
The probability that the range of the random sample of size 4 is less than 0.5 is \(\frac{7}{24}\).
Step by step solution
01
Understand the properties of uniform distribution and range
In a uniform distribution, all values have equal probabilities in the given interval. Here, \(f(x)=1,0<x<1\) which implies that any value between 0 and 1 has equal probability. Range is the difference between the maximum and minimum values in a dataset. In this case, the range of the random sample of size 4 should be less than 0.5.
02
Find the range and calculate the probability
Since each sample have uniform distribution over (0, 1), the minimum value of the sample would be near 0 and the maximum value would be near 1. Because probability would be derived from the area under the curve in the respective interval, and we are looking for the range to be less than 0.5, the area would be captured between x=0 and x=0.5. Now the task is to find the probability \(P(X_{max} - X_{min} < 0.5)\) where \(X_{max}\) and \(X_{min}\) are the maximum and minimum values of the sample respectively. The probability can be calculated by integrating the values of the probability density function (which is 1 here) over the feasible range. The feasible area for max value is confined by the range (i.e., max - min < 0.5)
03
Integrate over the feasible range
The feasible range of the maximum value (X_{max}) is between the minimum value (X_{min}) and \(X_{min} + 0.5\). So, integrate the pdf over this range for all possible values of minorities, meaning from 0 to 0.5 for minorities, and then for each value of min from min to min + 0.5 for max. As a result, the integral is \(\int_{0}^{0.5} dx_{min} \int_{x_{min}}^{x_{min} + 0.5} dx_{max}\). However this integral only covers half of the feasible area where \(x_{min}\) is less than 0.5 and therefore should be multiplied by 2 to cover the other half. Final expression becomes:\[2* \int_{0}^{0.5} dx_{min} \int_{x_{min}}^{x_{min} + 0.5} dx_{max}\]
04
Calculate the Integral
Calculate the integral to get the probability. Computing this integral you get \(\frac{7}{24}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Random Sample
Understanding random samples is crucial in statistics as they are representative of larger populations. A random sample refers to a set of individuals or items chosen from a larger pool where each member is selected by chance. By ensuring that each possible sample has an equal chance of being chosen, statisticians can make accurate inferences about the entire population.
Our exercise involves finding the probability of an event related to a random sample of size 4 from a uniform distribution. It's important to recognize the significance of the sample size and randomness. A larger sample size generally provides a better representation of the population, leading to more reliable results. However, in this instance, we are focused solely on a sample of four observations, making each selection's randomness critical for the integrity of our conclusions.
Random sampling is the foundation for creating datasets that avoid bias. In our exercise, we assume a perfectly random mechanism for picking the sample, which is practically challenging but theoretically ideal for probability calculations.
Our exercise involves finding the probability of an event related to a random sample of size 4 from a uniform distribution. It's important to recognize the significance of the sample size and randomness. A larger sample size generally provides a better representation of the population, leading to more reliable results. However, in this instance, we are focused solely on a sample of four observations, making each selection's randomness critical for the integrity of our conclusions.
Random sampling is the foundation for creating datasets that avoid bias. In our exercise, we assume a perfectly random mechanism for picking the sample, which is practically challenging but theoretically ideal for probability calculations.
Probability Density Function (PDF)
The probability density function (PDF), often simply referred to as the density of a continuous random variable, is an essential function in understanding probabilities. It describes the relative likelihood for a random variable to take on a given value. In the realm of continuous variables, the pdf helps us comprehend how values are distributed over an interval.
The pdf for a uniform distribution, like in our exercise, is deceptively simple: it assigns constant probability to each value in its interval. For the distribution given by \(f(x)=1, 0
When dealing with pdfs, the total area under the curve within the interval of possible values always equals 1, representing the fact that some outcome in the interval will certainly occur. Calculating probabilities then involves finding the area under the curve for specific ranges, which introduces us to the application of integral calculus in probability.
The pdf for a uniform distribution, like in our exercise, is deceptively simple: it assigns constant probability to each value in its interval. For the distribution given by \(f(x)=1, 0
When dealing with pdfs, the total area under the curve within the interval of possible values always equals 1, representing the fact that some outcome in the interval will certainly occur. Calculating probabilities then involves finding the area under the curve for specific ranges, which introduces us to the application of integral calculus in probability.
Range of a Sample
In statistics, the range of a sample provides a simple measure of the dispersion among the values in the sample. It's found by subtracting the smallest value (minimum) from the largest value (maximum). Understanding the range is crucial when trying to grasp the spread and the scale of variability in a sample.
The problem presented poses a scenario where we are looking for the probability that the range of a random sample of size four is less than half. This is not an arbitrary question but a calculated exploration into the behavior of a sample taken from a uniform distribution. The uniformity denotes that all values within a certain interval are equally probable, and the knowledge of the range provides insights into how tightly or loosely those sample values are clustered.
Especially in uniform distributions, knowing how values can spread from the minimum to the maximum helps us appreciate the underlying certainty or uncertainty in outcomes. For instance, if most samples from a distribution tend to have a small range, this implies less variability and higher predictability.
The problem presented poses a scenario where we are looking for the probability that the range of a random sample of size four is less than half. This is not an arbitrary question but a calculated exploration into the behavior of a sample taken from a uniform distribution. The uniformity denotes that all values within a certain interval are equally probable, and the knowledge of the range provides insights into how tightly or loosely those sample values are clustered.
Especially in uniform distributions, knowing how values can spread from the minimum to the maximum helps us appreciate the underlying certainty or uncertainty in outcomes. For instance, if most samples from a distribution tend to have a small range, this implies less variability and higher predictability.
Integral Calculus in Probability
The use of integral calculus in probability is a powerful tool for solving problems related to continuous random variables. Integrals allow us to calculate the area under the curve of a probability density function, which corresponds to the probability of the variable falling within a certain interval.
In our exercise, we use integral calculus specifically to find the probability that the range of a randomly selected sample from a uniform distribution is below a certain value. This involves setting up a double integral because we need to consider the minimum and maximum values simultaneously in our calculations. The first integral sums probabilities across the possible minimum values, and the second one, nested within the first, accumulates the probabilities across the maximum values, given each minimum.
The resulting expression of this double integration represents the total probability of the range being less than \(0.5\). This complex calculation is beautifully simplified by the properties of the uniform distribution and integral calculus. The combination acts as a bridge, translating a theoretical question about probability into a concrete answer by measuring the area under the curve.
In our exercise, we use integral calculus specifically to find the probability that the range of a randomly selected sample from a uniform distribution is below a certain value. This involves setting up a double integral because we need to consider the minimum and maximum values simultaneously in our calculations. The first integral sums probabilities across the possible minimum values, and the second one, nested within the first, accumulates the probabilities across the maximum values, given each minimum.
The resulting expression of this double integration represents the total probability of the range being less than \(0.5\). This complex calculation is beautifully simplified by the properties of the uniform distribution and integral calculus. The combination acts as a bridge, translating a theoretical question about probability into a concrete answer by measuring the area under the curve.
