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Chapter 6: Problem 8

The random variable \(X\) follows a Poisson process with the given mean. Assuming \(\mu=7,\) compute (a) \(P(10)\) (b) \(P(X<10)\) (c) \(P(X \geq 10)\) (d) \(P(7 \leq X \leq 9)\)

Short Answer

Expert verified
(a) 0.0901, (b) 0.700, (c) 0.300, (d) 0.380

Step by step solution

01

Understand the Poisson distribution

The Poisson distribution is used to model the number of events that occur within a fixed interval of time or space. The probability of observing exactly k events is given by the formula: \[ P(X=k) = \frac{{e^{-\mu} \mu^k}}{{k!}} \]where \( \mu \) is the mean number of events.
02

Compute \( P(10) \)

Use the formula for the Poisson distribution with \( k=10 \) and \( \mu=7 \):\[ P(X=10) = \frac{{e^{-7} 7^{10}}}{{10!}} = \frac{{e^{-7} 282475249}}{{3628800}} \approx 0.0901 \]
03

Compute the CDF for \( P(X

Calculate the cumulative distribution function (CDF) for \( k < 10 \):\[ P(X<10) = \sum_{k=0}^{9} \frac{{e^{-7} 7^k}}{{k!}} \]By summing these probabilities, we get approximately 0.700. This can be computed using a Python programming language or using a Poisson distribution table.
04

Compute \( P(X \, \geq \, 10) \)

To find \( P(X \geq 10) \), use the complement rule:\[ P(X \geq 10) = 1 - P(X < 10) \]Using the previous result, we get:\[ P(X \geq 10) = 1 - 0.700 = 0.300 \]
05

Compute \( P(7 \leq X \leq 9) \)

Sum the probabilities from 7 through 9:\[ P(7 \leq X \leq 9) = P(X=7) + P(X=8) + P(X=9) \]Using the Poisson formula:\[ P(X=7) = \frac{{e^{-7} 7^7}}{{7!}} \approx 0.1490 \]\[ P(X=8) = \frac{{e^{-7} 7^8}}{{8!}} \approx 0.1303 \]\[ P(X=9) = \frac{{e^{-7} 7^9}}{{9!}} \approx 0.1011 \]Summing these probabilities, we get approximately 0.3804.

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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 branch of mathematics that deals with the analysis of random phenomena. It provides a mathematical framework to quantify the likelihood of events occurring.
Random variables are used to represent these events and can be discrete or continuous. In our exercise, we dealt with a discrete random variable, X, representing the number of events occurring in a fixed interval.
Probabilities are represented by a number between 0 and 1. A probability of 0 indicates an impossible event, while a probability of 1 denotes a certain event. By utilizing probability theory, we can calculate the likelihood of different outcomes for random variables, which is fundamental in fields ranging from statistics to engineering.
Cumulative Distribution Function
The cumulative distribution function (CDF) of a discrete random variable is the function that gives the probability that the variable is less than or equal to a certain value. It is defined as:
\[ F(k) = P(X \leq k) \]
The CDF is a useful tool as it helps in determining the probability of a random variable falling within a specified range. For instance, in our exercise, we calculated the CDF to find the probability that X is less than 10. By summing up the individual probabilities from 0 to 9, we determined that approximately 70% of the time, the number of events will be less than 10.
Complement Rule
The complement rule is a fundamental principle in probability theory that states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. Mathematically, it is expressed as:
\[ P(A^c) = 1 - P(A) \]
where \(A^c\) denotes the complement of event A.
In the given exercise, we used the complement rule to determine the probability of X being greater than or equal to 10. By calculating the complement of the probability of X being less than 10, we found that P(X \geq 10) is approximately 0.300. This rule simplifies calculations and helps in finding the probability of the complementary events.
Poisson Probability Formula
The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space. It is particularly useful for rare events. The Poisson probability formula is given by:
\[ P(X=k) = \frac{e^{-\mu} \mu^k}{k!} \]
where \(\mu\) is the average number of events (mean), k is the actual number of events, and e is the base of the natural logarithm (approximately equal to 2.71828).
In our provided exercise, we used the Poisson formula to compute the probability of different event counts. For example, to find P(X=10), we substituted \(\mu=7\) and \(k=10\), giving us a probability of approximately 0.0901. This is analogous to finding the chance of witnessing exactly 10 events when the average rate is 7 events.

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Most popular questions from this chapter

Some standardized tests, such as the SAT test, incorporate a penalty for wrong answers. For example, a multiple-choice question with five possible answers will have 1 point awarded for a correct answer and \(\frac{1}{4}\) point deducted for an incorrect answer. Questions left blank are worth 0 points. (a) Find the expected number of points received for a multiplechoice question with five possible answers when a student just guesses. (b) Explain why there is a deduction for wrong answers.

Officer Thompson of the Bay Ridge Police Department works the graveyard shift. He averages 4.5 calls per shift from his dispatcher. Assume the number of calls follows a Poisson distribution. Would it be unusual for Officer Thompson to get fewer than 2 calls in a shift?

In a recent poll, the Gallup Organization found that \(45 \%\) of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. (a) Find and interpret the probability that exactly 15 of those surveyed feel the state of morals is poor. (b) Find and interpret the probability that no more than 10 of those surveyed feel the state of morals is poor. (c) Find and interpret the probability that more than 16 of those surveyed feel the state of morals is poor. (d) Find and interpret the probability that 13 or 14 believe the state of morals is poor. (e) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values is poor in the United States? Why?

BlackJack is a popular casino game in which a player is dealt two cards where the value of the card corresponds to the number on the card, face cards are worth ten, and aces are worth either one or eleven. The object is to get as close to 21 as possible without going over and have cards whose value exceeds that of the dealer. A blackjack is an ace and a ten in two cards. It pays 1.5 times the bet. The dealer plays last and must draw a card with sixteen and hold with seventeen or more. The following distribution shows the winnings and probability for a \(\$ 20\) bet. In cases where the dealer and player have the same value, there is a tie (called a "push"). Source: "Examining a Gambler's Claims: Probabilistic Fact-Checking and Don Johnson's Extraordinary Winning Streak" by W.J. Hurley, Jack Brimberg, and Richard Kohar. Chance Vol. 27.1,2014 $$ \begin{array}{cc} \text { Winnings } & \text { Probability } \\ \hline 0 & 0.0982 \\ \hline \$ 30 & 0.0483 \\ \hline \$ 20 & 0.389275 \\ \hline-\$ 20 & 0.464225 \end{array} $$ (a) Compute and interpret the expected value of the game from the player's point of view. (b) Suppose over the course of one hour, a player can expect to be dealt about 40 hands. How much should a player expect to win or lose over the course of three hours?

An investment counselor calls with a hot stock tip. He believes that if the economy remains strong, the investment will result in a profit of \(\$ 50,000\). If the economy grows at a moderate pace, the investment will result in a profit of \(\$ 10,000 .\) However, if the economy goes into recession, the investment will result in a loss of \(\$ 50,000\). You contact an economist who believes there is a \(20 \%\) probability the economy will remain strong, a \(70 \%\) probability the economy will grow at a moderate pace, and a \(10 \%\) probability the economy will slip into recession. What is the expected profit from this investment?
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