91Ó°ÊÓ

Consider a collection \(A_{1}, \ldots, A_{k}\) of mutually exclusive and exhaustive events, and a random variable \(X\) whose distribution depends on which of the \(A_{i}\) 's occurs (e.g., a commuter might select one of three possible routes from home to work, with \(X\) representing the commute time). Let \(E\left(X \mid A_{i}\right)\) denote the expected value of \(X\) given that the event \(A_{i}\) occurs. Then it can be shown that \(E(X)=\) \(\Sigma E\left(X \mid A_{i}\right) \cdot P\left(A_{i}\right)\), the weighted average of the individual "conditional expectations" where the weights are the probabilities of the partitioning events. a. The expected duration of a voice call to a particular telephone number is 3 minutes, whereas the expected duration of a data call to that same number is 1 minute. If \(75 \%\) of all calls are voice calls, what is the expected duration of the next call? b. A deli sells three different types of chocolate chip cookies. The number of chocolate chips in a type \(i\) cookie has a Poisson distribution with parameter \(\lambda_{i}=i+1\) \((i=1,2,3)\). If \(20 \%\) of all customers purchasing a chocolate chip cookie select the first type, \(50 \%\) choose the second type, and the remaining \(30 \%\) opt for the third type, what is the expected number of chips in a cookie purchased by the next customer?

Step by step solution

01

Understanding the Problem

We have a collection of mutually exclusive and exhaustive events, and we need to find the expected value of a random variable based on these events. The formula given is \(E(X) = \sum E(X | A_i) \cdot P(A_i)\), where \(E(X | A_i)\) is the expected value of \(X\) given the event \(A_i\) occurs.

02

Part a: Identify Given Information

For the call duration problem, \(E(X | A_1)\), the expected duration of a voice call, is 3 minutes and \(E(X | A_2)\), the expected duration of a data call, is 1 minute. The probability of a voice call (\(P(A_1)\)) is 0.75, and thus the probability of a data call (\(P(A_2)\)) is 0.25.

03

Part a: Calculate Expected Duration of Next Call

Using \(E(X) = \sum E(X | A_i) \cdot P(A_i)\), compute \(E(X) = 3 \cdot 0.75 + 1 \cdot 0.25 = 2.25 + 0.25 = 2.5\). Thus, the expected duration of the next call is 2.5 minutes.

04

Part b: Identify Given Information

In the cookie example, the expected number of chips for type 1, 2, and 3 cookies are \(E(X | A_1) = 2\), \(E(X | A_2) = 3\), and \(E(X | A_3) = 4\) respectively, as the expectation for a Poisson distribution is its parameter \(\lambda_i\). The probabilities are \(P(A_1) = 0.2\), \(P(A_2) = 0.5\), and \(P(A_3) = 0.3\).

05

Part b: Calculate Expected Number of Chips

Utilizing \(E(X) = \sum E(X | A_i) \cdot P(A_i)\), evaluate \(E(X) = 2 \cdot 0.2 + 3 \cdot 0.5 + 4 \cdot 0.3 = 0.4 + 1.5 + 1.2 = 3.1\). Hence, the expected number of chips in the next cookie is 3.1.

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Key Concepts

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

Mutually Exclusive Events

Mutually exclusive events refer to situations where the occurrence of one event prevents the occurrence of any other event in a given set. In simpler terms, if one event happens, the others cannot. For example, in the case of tossing a coin, the result can either be heads or tails, but never both. These events help in simplifying the calculation of probabilities.

When dealing with mutually exclusive events, it's particularly important to understand that the probability of any two events happening at the same time is zero. This characteristic is what defines their exclusivity. In the context of our original exercise, we have events like choosing a voice call or a data call, or choosing among different cookie types - each choice represents a mutually exclusive event since a call can only be one type at a time, and a purchased cookie can only belong to one type.

Moreover, being exhaustive means covering all possible scenarios. When events are exhaustive, their probabilities sum up to 1, because they encompass all potential outcomes. For instance, if 75% of calls are voice calls, then the rest must be data calls, adding up to 100% of calls.

Probability

Probability is a way of quantifying the likelihood that a certain event will happen. It ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty. To understand probability in practical terms, think of what happens when you roll a six-sided die. The probability of getting, say, a three, is 1/6 because three is one of the six possible outcomes.

In the context of our exercise, calculating probabilities helps in determining expectations. We apply the probability to the expected outcomes, essentially weighting those expectations by how likely they are to happen. So when we say there's a 75% chance that a call is a voice call, we're assigning a probability of 0.75 to that particular event happening. This probability helps us compute conditional expectations, as those probabilities are used as weights in calculating the expected value of the random variable.

• Calculating probabilities of mutually exclusive events involves adding their individual probabilities, knowing the total is 1 because they are exhaustive.
• Probabilities help make informed predictions about future events.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses how many times an event is likely to occur within a specified period. It applies when each event is rare relative to the possible number of occurrences, like getting a specific number of call drops per hour or the number of cookies with exactly three chocolate chips sold in a day.

For events following a Poisson distribution, we mainly need one parameter: \( \lambda \) (lambda), which represents the average number of times that event occurs in a given interval. The expectation (average) of a Poisson-distributed random variable is exactly its parameter, \( E(X) = \lambda \).

In our exercise, each type of cookie has a number of chocolate chips distributed according to a Poisson distribution with its own \( \lambda \). Understanding this allows us to calculate an expected number of outcomes like the expected number of chocolate chips for any randomly chosen cookie. This expectation is fundamental for statistical predictions in situations with rare events as in the Poisson distribution.