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Chapter 5: Problem 17

Let \(X\) be a random variable with cumulative distribution function $$ F(x)=\left\\{\begin{array}{ll} 0, & \text { if } x<0 \\ \sin ^{2}(\pi x / 2), & \text { if } 0 \leq x \leq 1 \\ 1, & \text { if } 1

Short Answer

Expert verified
(a) \( f_X(x) = \frac{\pi}{2} \sin(\pi x) \) for \( 0 \leq x \leq 1 \). (b) \( P(X < 1/4) = \frac{2 - \sqrt{2}}{4} \).

Step by step solution

01

Determine the Interval for Derivative

To find the density function, we need to differentiate the cumulative distribution function (CDF) where it is continuous. The CDF given is piecewise, so we focus on the interval where it's not a constant, that is \(0 \leq x \leq 1\).
02

Differentiate the CDF

For the interval \(0 \leq x \leq 1\), the CDF is \( F(x) = \sin^2(\pi x / 2) \). We differentiate this with respect to \(x\) to find the probability density function (PDF).Using the chain rule:\[ f_X(x) = \frac{d}{dx} \left( \sin^2\left(\frac{\pi x}{2}\right) \right) = 2 \sin\left(\frac{\pi x}{2}\right) \cos\left(\frac{\pi x}{2}\right) \cdot \frac{\pi}{2} = \frac{\pi}{2} \sin(\pi x) \]
03

Verify the Density Function Condition

The PDF must integrate to 1 over its entire range to confirm it's a valid density function. Since \(\sin(\pi x)\) is a well-known function over its period, we verify if it meets the condition:\[ \int_0^1 \frac{\pi}{2} \sin(\pi x) \, dx = \left[ -\frac{1}{2}\cos(\pi x) \right]_0^1 = 1 \]This confirms that \(f_X(x)\) is a valid PDF.
04

Calculate the Probability for Interval Condition

For the probability that \( X < 1/4 \), we evaluate the CDF at \( x = 1/4 \):\[ P(X < 1/4) = F(1/4) = \sin^2\left(\frac{\pi}{2} \cdot \frac{1}{4}\right) = \sin^2\left(\frac{\pi}{8}\right) \]Since \( \sin^2(x) = (1 - \cos(2x))/2 \):\[ \sin^2\left(\frac{\pi}{8}\right) = \frac{1}{2} \left(1 - \cos\left(\frac{\pi}{4}\right)\right) = \frac{1}{2}\left(1 - \frac{\sqrt{2}}{2}\right) = \frac{2 - \sqrt{2}}{4} \]

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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 understanding the behavior of continuous random variables. It describes the likelihood of a random variable to take on a particular value, although it's important to note that the actual probability of a continuous random variable taking any exact value is zero.
Instead, PDFs help calculate the probability that a random variable falls within a certain range.
Visually, the PDF can be seen as the curve that describes how the variable's potential values are distributed. The area under this curve for a given interval represents the probability that the variable falls into that interval. Mathematically, if you integrate the PDF over the entire range of possible values, the result is always 1. This total area of 1 reflects the certainty that the random variable takes some value in the space it describes.
Random Variable
Random variables are a fundamental concept in statistics and probability theory. They essentially map outcomes of a random process to numerical values, allowing us to apply mathematical techniques to these outcomes.
There are two main types of random variables: discrete and continuous.
  • Discrete Random Variables: These take on a countable number of distinct values. For instance, rolling a six-sided die results in a discrete random variable with values 1 through 6.
  • Continuous Random Variables: These can take any value within a range. For example, measuring the height of students in a class represents a continuous random variable.
The cumulative distribution function (CDF) you encountered in the exercise maps cumulative probabilities up to any given point for a random variable. It helps us see how probabilities accumulate and is a stepping stone to finding the PDF.
Chain Rule Differentiation
In calculus, the chain rule is an essential technique for differentiating composite functions. It helps us deal with functions nested within one another by considering how changes in the outer function relate to changes in the inner function.
The polynomial \(F(x) = \sin^{2}(\pi x / 2)\), from your exercise, is a perfect example of applying the chain rule.
How does it work?
  • Identify the outer function. In the example, \(\phi(u) = u^2\) is the outer function.
  • Identify the inner function. Here, \(u = \sin(\pi x / 2)\).
  • Differentiate each function separately. First, find \(\phi'(u) = 2u\) for the outer function. Then, \(\frac{d}{du}(\sin(\pi x / 2)) = \frac{\pi}{2} \cos(\pi x / 2)\) for the inner function.
  • Apply the chain rule: \(\frac{d}{dx} \phi(u(x)) = \phi'(u) \times \frac{du}{dx} = 2\sin(\pi x / 2) \times \frac{\pi}{2} \cos(\pi x / 2)\)
Chain rule differentiation helps compute the correct derivatives crucial in finding the PDF from the CDF, as seen in the exercise.

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Most popular questions from this chapter

A census in the United States is an attempt to count everyone in the country. It is inevitable that many people are not counted. The U. S. Census Bureau proposed a way to estimate the number of people who were not counted by the latest census. Their proposal was as follows: In a given locality, let \(N\) denote the actual number of people who live there. Assume that the census counted \(n_{1}\) people living in this area. Now, another census was taken in the locality, and \(n_{2}\) people were counted. In addition, \(n_{12}\) people were counted both times. (a) Given \(N, n_{1},\) and \(n_{2},\) let \(X\) denote the number of people counted both times. Find the probability that \(X=k,\) where \(k\) is a fixed positive integer between 0 and \(n_{2}\). (b) Now assume that \(X=n_{12}\). Find the value of \(N\) which maximizes the expression in part (a). Hint: Consider the ratio of the expressions for successive values of \(N\).

Suppose we are attending a college which has 3000 students. We wish to choose a subset of size 100 from the student body. Let \(X\) represent the subset, chosen using the following possible strategies. For which strategies would it be appropriate to assign the uniform distribution to \(X ?\) If it is appropriate, what probability should we assign to each outcome? (a) Take the first 100 students who enter the cafeteria to eat lunch. (b) Ask the Registrar to sort the students by their Social Security number, and then take the first 100 in the resulting list. (c) Ask the Registrar for a set of cards, with each card containing the name of exactly one student, and with each student appearing on exactly one card. Throw the cards out of a third-story window, then walk outside and pick up the first 100 cards that you find.

Write a computer algorithm that simulates a hypergeometric random variable with parameters \(N, k,\) and \(n\).

Suppose we are observing a process such that the time between occurrences is exponentially distributed with \(\lambda=1 / 30\) (i.e., the average time between occurrences is 30 minutes). Suppose that the process starts at a certain time and we start observing the process 3 hours later. Write a program to simulate this process. Let \(T\) denote the length of time that we have to wait, after we start our observation, for an occurrence. Have your program keep track of \(T\). What is an estimate for the average value of \(T ?\)

In a class of 80 students, the professor calls on 1 student chosen at random for a recitation in each class period. There are 32 class periods in a term. (a) Write a formula for the exact probability that a given student is called upon \(j\) times during the term. (b) Write a formula for the Poisson approximation for this probability. Using your formula estimate the probability that a given student is called upon more than twice.
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