Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 16: Problem 6
a If \(X\) is the random variable which follows the binomial distribution with \(n=600\) and \(p=1 \times 10^{-3}\), evaluate (to 4 d.p.) i \(P(X=1)\) ii \(P(X=2)\) iii \(P(X=3)\) The Poisson distribution is described by $$ P(X=x)=\frac{e^{-n p}(n p)^{x}}{\chi !} $$ b For this distribution, evaluate (consider the above values of \(n\) and \(p\) ) i \(P(X=1)\) ii \(P(X=2)\) iii \(P(X=3)\) c What do you notice about the results of a and \(\mathbf{b}\) ?
Short Answer
Expert verified
Binomial: 0.3288, 0.0986, 0.0197. Poisson: 0.3293, 0.0988, 0.0198. Values are very similar.
Step by step solution
01
Understand the Binomial Distribution
The binomial distribution is given by the formula \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \(n\) is the number of trials, \(p\) is the probability of success, and \(k\) is the number of successes.
02
Calculate \(P(X=1)\) for Binomial Distribution
Using the binomial distribution formula: \[P(X=1) = \binom{600}{1}(1 \times 10^{-3})^1 (1 - 1 \times 10^{-3})^{600-1}\]Calculate the combination and the powers:\( \binom{600}{1} = 600\)\( (1 \times 10^{-3})^1 = 1 \times 10^{-3} \)\( (1 - 1 \times 10^{-3})^{599} \approx e^{-0.6} \approx 0.5488 \)Multiplying these together:\[ P(X=1) = 600 \times 1 \times 10^{-3} \times 0.5488 = 0.3288 \]
03
Calculate \(P(X=2)\) for Binomial Distribution
Using the binomial distribution formula:\[ P(X=2) = \binom{600}{2}(1 \times 10^{-3})^2 (1 - 1 \times 10^{-3})^{598} \]Calculate the combination and the powers:\( \binom{600}{2} = \frac{600 \times 599}{2 \times 1} = 179700 \)\( (1 \times 10^{-3})^2 = 10^{-6} \)\( (1 - 1 \times 10^{-3})^{598} \approx e^{-0.6} \approx 0.5488 \)Multiplying these together:\[P(X=2) = 179700 \times 10^{-6} \times 0.5488 = 0.0986 \]
04
Calculate \(P(X=3)\) for Binomial Distribution
Using the binomial distribution formula:\[ P(X=3) = \binom{600}{3}(1 \times 10^{-3})^3 (1 - 1 \times 10^{-3})^{597} \]Calculate the combination and the powers:\( \binom{600}{3} = \frac{600 \times 599 \times 598}{3 \times 2 \times 1} = 35979900 \)\( (1 \times 10^{-3})^3 = 10^{-9} \)\( (1 - 1 \times 10^{-3})^{597} \approx e^{-0.6} \approx 0.5488 \)Multiplying these together:\[ P(X=3) = 35979900 \times 10^{-9} \times 0.5488 = 0.0197 \]
05
Understand the Poisson Distribution
The Poisson distribution is an approximation of the binomial distribution when \(n\) is large and \(p\) is small. The Poisson distribution is given by:\[ P(X=x) = \frac{e^{-np} (np)^x}{x!} \] where \( \text{mean} = np \). Here, \( \text{mean} = 600 \times 10^{-3} = 0.6 \).
06
Calculate \(P(X=1)\) for Poisson Distribution
Using the Poisson distribution formula:\[ P(X=1) = \frac{e^{-0.6}(0.6)^1}{1!} = e^{-0.6} \times 0.6 \approx 0.5488 \times 0.6 = 0.3293 \]
07
Calculate \(P(X=2)\) for Poisson Distribution
Using the Poisson distribution formula:\[ P(X=2) = \frac{e^{-0.6}(0.6)^2}{2!} = e^{-0.6} \times 0.36 / 2 \approx 0.5488 \times 0.18 = 0.0988 \]
08
Calculate \(P(X=3)\) for Poisson Distribution
Using the Poisson distribution formula:\[ P(X=3) = \frac{e^{-0.6}(0.6)^3}{3!} = e^{-0.6} \times 0.216 / 6 \approx 0.5488 \times 0.036 = 0.0198 \]
09
Compare the Results
The results from steps 2, 3, and 4 (binomial distribution) are 0.3288, 0.0986, and 0.0197 respectively, while the results from steps 6, 7, and 8 (Poisson distribution) are 0.3293, 0.0988, and 0.0198 respectively. The values are quite close, showing that the Poisson distribution is a good approximation for the binomial distribution in this case.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Distribution
The binomial distribution is a probability distribution that summarizes the likelihood of a value taking one of two independent states. For example, success or failure. It is defined by the formula \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Here, \(n\) is the number of trials, \(p\) is the probability of success in a single trial, and \(k\) is the number of successes. To calculate probabilities: - Determine the number of trials (\(n\)) and probability of success (\(p\)). - Use the formula to find the probability for the required number of successes (\(k\)). This distribution is commonly used in scenarios like quality control in manufacturing, where the interest is in the number of defective items in a batch.
Poisson Distribution
The Poisson distribution is useful when you are dealing with the number of events occurring within a fixed interval of time or space. It's defined as \[P(X=x)=\frac{e^{-np}(np)^{x}}{\text{x!}} \] Here, the mean of the distribution is \(np\), where \(n\) is the number of trials, and \(p\) is the probability of success in a single trial. To use the Poisson distribution: - Calculate the mean (\(np\)). - Substitute the mean and \(x\) (number of events) into the formula. The Poisson distribution is often applied in fields such as telecommunication (e.g., the number of calls received at a call center in an hour). It serves as a good approximation for the binomial distribution when \(n\) is large and \(p\) is small.
Probability Calculations
Probability calculations are crucial for assessing the likelihood of various outcomes. Using the binomial and Poisson formulas, we can calculate specific probabilities: - For the binomial distribution, we multiply the combination of successes, the probability raised to the success count, and the probability of failure raised to the remaining trials. - For the Poisson distribution, we multiply the exponential of the negative mean, the mean raised by the success number, divided by the factorial of the number of successes. Understanding these calculations helps in evaluating scenarios and making informed predictions about the frequency of events.
Approximations
Approximation methods, like using the Poisson distribution to approximate the binomial distribution, simplify complex calculations. The conditions for such an approximation are: - \(n\) should be large. - \(p\) should be small. When these conditions are met, the Poisson distribution provides a good approximation of the binomial distribution. The comparison between binomial and Poisson calculations in the exercise shows negligible differences between them. This indicates how effective Poisson distribution can be in approximating binomial probabilities in suitable scenarios.
