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Chapter 3: Problem 5

Suppose that \(\left\\{E_{n}, n \geq 1\right\\}\) and \(\left\\{F_{n}, n \geq 1\right\\}\) are increasing sequences of events having limits \(E\) and \(F\). Show that if \(E_{n}\) is independent of \(F_{n}\) for all \(n\), then \(E\) is independent of \(F\).

Short Answer

Expert verified
We are given that the sequences of events \(\{E_n\}\) and \(\{F_n\}\) are increasing and have limits \(E\) and \(F\), respectively. It is also given that \(E_n\) is independent of \(F_n\) for all \(n\). In order to show that the limit events \(E\) and \(F\) are independent, we need to demonstrate that \(P(E \cap F) = P(E)P(F)\). Using the given independence and continuous property of probability measures, we find that \(P(E \cap F) = P\left(\bigcap_{n = 1}^\infty E_n \cap \bigcap_{n = 1}^\infty F_n\right) = \lim_{n \rightarrow \infty} P(E_n)P(F_n) = P(E)P(F)\). Therefore, we have proved that if \(E_n\) is independent of \(F_n\) for all \(n\), then \(E\) is independent of \(F\).

Step by step solution

01

Define the given sequences

We are given that \(\left\\{E_{n}, n \geq 1\right\\}\) and $\left\\{F_{n}, n \geq 1\right\\}\( are increasing sequences of events having limits \)E\( and \)F\(. That means for all \)n \geq 1$, $$E_1 \subseteq E_2 \subseteq E_3 \subseteq \cdots$$ and, $$F_1 \subseteq F_2 \subseteq F_3 \subseteq \cdots$$ with \(\lim_{n \rightarrow \infty} E_n = E\) and \(\lim_{n \rightarrow \infty} F_n = F\).
02

Define independence of events

Two events \(A\) and \(B\) are independent if and only if: $$P(A \cap B) = P(A)P(B)$$ In our case, we are given that \(E_n\) is independent of \(F_n\) for all \(n\). So, we have: $$P(E_n \cap F_n) = P(E_n)P(F_n)$$ for all \(n \geq 1\).
03

Show the independence of the limits

Our goal is to show that the limits \(E\) and \(F\) are also independent, that is: $$P(E \cap F) = P(E)P(F)$$ To show this, we will use the following identities: $$P(E) = \lim_{n \rightarrow \infty} P(E_n)$$ $$P(F) = \lim_{m \rightarrow \infty} P(F_m)$$ We will also use the fact that if \(A \subseteq B\), then \(P(A) \leq P(B)\). Now, consider the following sequence of probabilities: $$P(E_n \cap F_n) = P(E_n)P(F_n)$$ $$P\left(\bigcap_{n = 1}^\infty (E_n \cap F_n)\right) = \lim_{n \rightarrow \infty} P(E_n \cap F_n) = \lim_{n \rightarrow \infty} P(E_n)P(F_n)$$ Applying the continuous property of probability measures: $$P\left(\bigcap_{n = 1}^\infty (E_n \cap F_n)\right) = P\left(\bigcap_{n = 1}^\infty E_n \cap \bigcap_{n = 1}^\infty F_n\right)$$ As the limits of the sequences are \(E\) and \(F\), we have: $$P(E \cap F) = P\left(\bigcap_{n = 1}^\infty E_n \cap \bigcap_{n = 1}^\infty F_n\right) = \lim_{n \rightarrow \infty} P(E_n)P(F_n) = P\left(\lim_{n\rightarrow \infty} E_n\right) P\left(\lim_{n\rightarrow \infty} F_n\right) = P(E)P(F)$$ Hence, we have proved that if \(E_n\) is independent of \(F_n\) for all \(n\), then \(E\) is independent of \(F\).

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Key Concepts

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

Probability Theory
Probability theory is the mathematical framework for dealing with experiments and phenomena that have uncertain outcomes. It provides tools for describing and predicting the likelihood of events. At its core are random variables, probability spaces, and events.

A probability space is a mathematical construct that models a random experiment, consisting of a sample space (all possible outcomes), a set of events (possible outcomes or combinations of outcomes), and a probability measure that assigns a probability to each event.

Understanding how to calculate probabilities and how different events interact is foundational for working with more advanced concepts such as the independence of events, a principle that has a crucial role in probability theory.
Limit of Events
In probability theory, the limit of a sequence of events offers insight into the behavior of these events as we proceed indefinitely. It is a concept drawn from calculus and analysis, where the limit of a function or sequence outside probability theory describes the value it approaches as the input or index grows without bound.

For a sequence of events, the notion of a limit is applied in a similar way. Specifically, if we have an increasing sequence of events鈥攎eaning each event is contained within the next鈥攖he limit is the event that encompasses all possible outcomes that occur infinitely often in the sequence. It is the 'destination' of the sequence, the ultimate event to which all the events in the sequence are leading.
Independent Events
Independent events in probability are those whose occurrence does not affect the likelihood of the other events' occurrences. Formally, two events, A and B, are independent if the probability of their intersection equals the product of their individual probabilities, denoted by the formula:
\[ P(A \cap B) = P(A)P(B) \]
Independence is a fundamental concept as it simplifies the computation of probabilities in complex situations. It is crucial to distinguish between independent and mutually exclusive events; the latter can't occur simultaneously, unlike independent events, which can occur at the same time without influencing each other.
Sequences of Events
Sequences of events refer to a series of events occurring in a specific order. In probability, we often deal with sequences to track the evolution of events over time, similar to monitoring a process or experiment as it progresses. An increasing sequence of events, for example, could represent a scenario where each event includes or is a superset of the one before.

Understanding sequences of events is critical when considering long-term trends or behaviors in probabilistic systems. It aids in comprehending how probabilities might converge over time, and it allows for the utilization of convergence theorems that provide valuable results in the realm of probability and statistics.

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Most popular questions from this chapter

English and American spellings are rigour and rigor, respectively. A man staying at a Parisian hotel writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40 percent of the English- speaking men at the hotel are English and 60 percent are Americans, what is the probability that the writer is an Englishman?

Consider two independent tosses of a fair coin. Let \(A\) be the event that the first toss lands heads, let \(B\) be the event that the second toss lands heads, and let \(C\) be the event that both land on the same side. Show that the events \(A\), \(B, C\) are pairwise independent-that is, \(A\) and \(B\) are independent, \(A\) and \(C\) are independent, and \(B\) and \(C\) are independent-but not independent.

There is a 50 - 50 chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia. If the queen has had three princes without the disease, what is the probability the queen is a carrier? If there is a fourth prince, what is the probability that he will have hemophilia?

What is the probability that at least one of a pair of fair dice lands on 6, given that the sum of the dice is \(i, i=2,3, \ldots, 12 ?\)

A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene in the sense that if an organism possesses the gene pair \(x X\), then it will outwardly have the appearance of the \(X\) gene. For instance, if \(X\) stands for brown eyes and \(x\) for blue eyes, then an individual having either gene pair \(X X\) or \(x X\) will have brown eyes, whereas one having gene pair \(x x\) will have blue eyes. The characteristic appearance of an organism is called its phenotype, whereas its genetic constitution is called its genotype. (Thus 2 organisms with respective genotypes \(a A, b B, c c\), \(d D, e e\) and \(A A, B B, c c, D D, e e\) would have different genotypes but the same phenotype.) In a mating between 2 organisms each one contributes, at random, one of its gene pairs of each type. The 5 contributions of an organism (one of each of the 5 types) are assumed to be independent and are also independent of the contributions of its mate. In a mating between organisms having genotypes \(a A, b B, c C, d D, e E\) and \(a a, b B, c c, D d, e e\) what is the probability that the progeny will (i) phenotypically and (ii) genotypically resemble (a) the first parent; (b) the second parent; (c) either parent; (d) neither parent?
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