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Chapter 1: Problem 35

Let the space of the random variable \(X\) be \(\mathscr{A}=\\{x ; 0

Short Answer

Expert verified
The probability of \(A_{2}\), \(P(A_{2})\), is \(3/4\).

Step by step solution

01

Understand the Given Data

First, understand the variables and their given values. Here, the probability space for \(X\) is defined from 0 to 1 exclusive, and two partitions, \(A_{1}\) and \(A_{2}\), are given. Also, the probability of \(A_{1}\), \(P(A_{1})\), is provided as \(1/4\).
02

Calculate \(P(A_{2})\)

The sum of the probabilities of all possible outcomes is 1. As \(A_{1}\) and \(A_{2}\) are the only subsets in the sample space, the sum of \(P(A_{1})\) and \(P(A_{2})\) should be equal to 1 i.e., \(P(A_{1}) + P(A_{2}) = 1\). Given that \(P(A_{1}) = 1/4\), we can substitute this value in the earlier equation to calculate \(P(A_{2})\), which is therefore \(P(A_{2}) = 1 - P(A_{1}) = 1 - 1/4 = 3/4\).

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Key Concepts

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

Probability Space
In probability theory, understanding the concept of a probability space is crucial. A probability space is the mathematical framework within which we study random processes and events.A probability space consists of three main components:
  • A sample space (\(\Omega\)) - This is the set of all possible outcomes of a random experiment. It represents the universe of potential results. In our exercise, the sample space is \(\mathscr{A}=\{x ; 0
  • Events - These are specific outcomes or combinations of outcomes. Each event is a subset of the sample space. In our given problem, \(A_{1}\) and \(A_{2}\) are examples of events.
  • A probability measure (\(P\)) - This assigns a probability to each event in the sample space. It is a function that maps events to a probability value between 0 and 1, such that the probability of the entire sample space is 1.
Each of these components interacts to help us calculate probabilities of various events, just like finding the probability of \(A_{2}\) given \(A_{1}\)'s probability.
Random Variable
A random variable is a function that assigns a numerical value to each outcome of a random experiment. It translates the uncertainty of an event into a quantitative representation.There are two main types of random variables:
  • Discrete Random Variables - These can take on a countable number of distinct values. Examples include the roll of a die or number of heads in a series of coin tosses.
  • Continuous Random Variables - These can take on a continuous range of values, often represented as intervals on the real number line. In the exercise, \(X\) is a continuous random variable with values over the interval \(0
Random variables are essential for calculating expectations, variances, and other statistical measures, representing real-world uncertainties in a manageable and interpretable form.
Conditional Probability
Conditional probability deals with finding the probability of an event given that another event has already occurred. It adjusts the probability based on new information that might alter the outcome.The formula for conditional probability is given by:\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]Where:
  • \(P(A|B)\) is the probability of event \(A\) occurring given that \(B\) has occurred.
  • \(P(A \cap B)\) is the probability of both events \(A\) and \(B\) occurring simultaneously.
  • \(P(B)\) is the probability of event \(B\).
In many real-world instances, conditional probability helps us make informed predictions by incorporating relevant information, just like how understanding \(P(A_{1})\) helps us determine \(P(A_{2})\) in related probability scenarios.

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Most popular questions from this chapter

Show that the function \(F(x, y)\) that is equal to 1, provided \(x+2 y \geq 1\), and that is equal to zero provided \(x+2 y<1\), cannot be a distribution function of two random variables. Hint. Find four numbers \(a

Let \(X\) be a random variable such that \(E\left[(X-b)^{2}\right]\) exists for all real \(b\). Show that \(E\left[(X-b)^{2}\right]\) is a minimum when \(b=E(X)\).

If \(C_{1}, C_{2}\), and \(C_{3}\) are subsets of \(\mathscr{C}\), show that $$\begin{aligned}P\left(C_{1} \cup C_{2} \cup C_{3}\right)=P\left(C_{1}\right) &+P\left(C_{2}\right)+P\left(C_{3}\right)-P\left(C_{1} \cap C_{2}\right) \\ &-P\left(C_{1} \cap C_{3}\right)-P\left(C_{2} \cap C_{3}\right)+P\left(C_{1} \cap C_{2} \cap C_{3}\right)\end{aligned}$$ What is the generalization of this result to four or more subsets of \(\mathscr{C}\) ?

Three distinct integers are chosen at random from the first 20 positive integers. Compute the probability that: (a) their sum is even; (b) their product is even.

Let \(X\) have the uniform distribution given by the p.d.f. \(f(x)=\frac{1}{5}\). \(x=-2,-1,0,1,2\), zero elsewhere. Find the p.d.f. of \(Y=X^{2} .\) Hint. Note that \(Y\) has a distribution of the discrete type.
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