Chapter 1: Problem 10
Let \(0
Short Answer
Expert verified
The 20th percentile of the distribution that has the pdf \(f(x)=4 x^{3}\) in the range \(0<x<1\) is approximately \(0.67\).
Step by step solution
01
Compute the Cumulative Distribution Function
The cumulative distribution function (CDF) is the integral of the probability density function from the lower limit to \(x\), in this case from 0 to \(x\). Thus, \(F(x) = \int_{0}^{x} f(t) dt = \int_{0}^{x} 4t^{3} dt\), which results in \(F(x) = x^{4}\).
02
Setting the Cumulative Distribution Function equal to the Percentile
Now, to find the 20th percentile (\(\xi_{0.20}\)), set \(F(x) = 0.20\) and solve for \(x\). In other words, find \(x\) such that \(x^{4} = 0.20\).
03
Solving for \(x\)
To isolate \(x\), take both sides of the equation to the 1/4 power (also known as the fourth root). \(x = (0.20)^{1/4} = 0.67 (rounded to two decimal places)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cumulative Distribution Function
The cumulative distribution function (CDF) plays a crucial role in understanding the behavior of random variables in statistics. It helps us determine the probability that a random variable takes a value less than or equal to a specific point. To grasp this concept, imagine the CDF as a running total of probabilities.
- The CDF, denoted as \( F(x) \), is calculated by integrating the probability density function (PDF) from the lowest value up to \( x \).
- In the context of continuous random variables, the CDF is a continuous and non-decreasing function.
- It always ranges from 0 to 1, starting at 0 as \( x \) begins from the lower bound and reaching 1 as \( x \) approaches the upper bound.
Probability Density Function
The probability density function (PDF) is a key element in understanding how probability is distributed across the possible values of a continuous random variable. Unlike discrete random variables, which have specific probabilities for each outcome, continuous random variables are described through densities.
- The PDF, \( f(x) \), describes how densely packed the probabilities are in a certain region of the possible outcomes.
- This function is not a probability itself but rather reflects how concentrated the probability is around different values of \( x \).
- The area under the PDF curve across an interval gives the probability that the variable falls within that interval.
Continuous Random Variables
Continuous random variables represent quantities that can take any value within a given interval. Unlike discrete variables, which have distinct and separate values, continuous variables can have an infinite number of possibilities.
- These variables are described by distributions represented by functions like the probability density function (PDF), which illustrate how probabilities are spread over the range of possible outcomes.
- They are often used in real-world scenarios where precision is vital, such as measuring time, height, or temperature.
- A key property is that the probability of the variable attaining any exact value is essentially zero because of the infinite possible values. Instead, we consider probabilities over intervals.