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Magazine Chapter 4: Problem 58
Compare the Poisson approximation with the correct binomial probability for the following cases: (a) \(P\\{X=2\\}\) when \(n=8, p=.1\); (b) \(P\\{X=9\\}\) when \(n=10, p=.95\); (c) \(P\\{X=0\\}\) when \(n=10, p=.1\); (d) \(P\\{X=4\\}\) when \(n=9, p=.2\).
Short Answer
Expert verified
The comparisons between the binomial probabilities and Poisson approximations are:
(a) \(P(X=2) \approx 0.2335\) (binomial) and \(0.2294\) (Poisson);
(b) \(P(X=9) \approx 0.3158\) (binomial) and \(0.3233\) (Poisson);
(c) \(P(X=0) \approx 0.3487\) (binomial) and \(0.3679\) (Poisson);
(d) \(P(X=4) \approx 0.1633\) (binomial) and \(0.1745\) (Poisson).
Step by step solution
01
(a) Calculate binomial probability when n=8 and p=.1
First, we calculate the binomial probability for \(X=2\).
\[P(X=2) = \binom{8}{2} (0.1)^2 (1-0.1)^{8-2} = \frac{8!}{2!(8-2)!} (0.1)^2 (0.9)^6\]
After simplifying, we find that \(P(X=2) \approx 0.2335\).
02
(a) Calculate Poisson approximation for n=8 and p=.1
Now, we calculate the Poisson approximation for the same case.
\[P(X=2) \approx e^{-8(0.1)}\frac{(8(0.1))^2}{2!} = e^{-0.8} \frac{(0.8)^2}{2}\]
After simplifying, we find that \(P(X=2) \approx 0.2294\).
03
(b) Calculate binomial probability when n=10 and p=.95
First, calculate the binomial probability for \(X=9\).
\[P(X=9) = \binom{10}{9} (0.95)^9 (1-0.95)^{10-9} = \frac{10!}{9!(10-9)!} (0.95)^9 (0.05)^1\]
After simplifying, we find that \(P(X=9) \approx 0.3158\).
04
(b) Calculate Poisson approximation for n=10 and p=.95
Now, we calculate the Poisson approximation for the same case.
\[P(X=9) \approx e^{-10(0.95)}\frac{(10(0.95))^9}{9!} = e^{-9.5} \frac{(9.5)^9}{9!}\]
After simplifying, we find that \(P(X=9) \approx 0.3233\).
05
(c) Calculate binomial probability when n=10 and p=.1
First, calculate the binomial probability for \(X=0\).
\[P(X=0) = \binom{10}{0} (0.1)^0 (1-0.1)^{10-0} = \frac{10!}{0!(10-0)!} (0.1)^0 (0.9)^{10}\]
After simplifying, we find that \(P(X=0) \approx 0.3487\).
06
(c) Calculate Poisson approximation for n=10 and p=.1
Now, we calculate the Poisson approximation for the same case.
\[P(X=0) \approx e^{-10(0.1)}\frac{(10(0.1))^0}{0!} = e^{-1} \frac{(1)^0}{1}\]
After simplifying, we find that \(P(X=0) \approx 0.3679\).
07
(d) Calculate binomial probability when n=9 and p=.2
First, calculate the binomial probability for \(X=4\).
\[P(X=4) = \binom{9}{4} (0.2)^4 (1-0.2)^{9-4} = \frac{9!}{4!(9-4)!} (0.2)^4 (0.8)^5\]
After simplifying, we find that \(P(X=4) \approx 0.1633\).
08
(d) Calculate Poisson approximation for n=9 and p=.2
Now, we calculate the Poisson approximation for the same case.
\[P(X=4) \approx e^{-9(0.2)}\frac{(9(0.2))^4}{4!} = e^{-1.8} \frac{(1.8)^4}{24}\]
After simplifying, we find that \(P(X=4) \approx 0.1745\).
To summarize the comparisons between the binomial probabilities and Poisson approximations:
(a) \(P(X=2) \approx 0.2335\) (binomial) and \(0.2294\) (Poisson);
(b) \(P(X=9) \approx 0.3158\) (binomial) and \(0.3233\) (Poisson);
(c) \(P(X=0) \approx 0.3487\) (binomial) and \(0.3679\) (Poisson);
(d) \(P(X=4) \approx 0.1633\) (binomial) and \(0.1745\) (Poisson).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Distribution
The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent experiments. Each of these experiments is also known as a Bernoulli trial. In a Bernoulli trial, there are only two possible outcomes: success or failure.
When you see the formula for the binomial probability, it captures how likely it is to get exactly 'k' successes in 'n' trials. The probability of success in each trial is denoted by 'p'.
For example, when calculating the probability of obtaining two heads in eight coin flips with a probability 'p' of landing heads, we use the binomial distribution.
When you see the formula for the binomial probability, it captures how likely it is to get exactly 'k' successes in 'n' trials. The probability of success in each trial is denoted by 'p'.
- The formula is: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\], where \(\binom{n}{k}\) is a binomial coefficient indicating the number of ways 'k' successes can occur in 'n' trials.
- This distribution is particularly useful when you are dealing with a finite number of trials and each trial is independent.
For example, when calculating the probability of obtaining two heads in eight coin flips with a probability 'p' of landing heads, we use the binomial distribution.
Probability Calculation
Probability calculation is key in understanding how likely an event is to occur. In probability, events are expressed numerically between 0 and 1鈥�0 meaning impossible and 1 meaning certain.
Moreover, when working with formulas derived from distributions like binomial or Poisson, these principles help break down how these probabilities are calculated. Knowing how to handle combinations (as in the binomial case) and understanding exponential functions (as in the Poisson case) ultimately allow for accurate probability determination.
- The first step in calculating the probability is identifying the total number of possible outcomes. This is crucial especially when dealing with real-world events.
- Next is finding the number of successful outcomes or 'favorable' outcomes for that event.
- The simple probability formula is: \[Probability = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\]
Moreover, when working with formulas derived from distributions like binomial or Poisson, these principles help break down how these probabilities are calculated. Knowing how to handle combinations (as in the binomial case) and understanding exponential functions (as in the Poisson case) ultimately allow for accurate probability determination.
Poisson Distribution
The Poisson distribution is another type of discrete probability distribution. It is especially useful as an approximation to the binomial distribution when 'n' is large and 'p' is small. The Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time or space.
This distribution is typically used when calculating the probability of a certain number of events happening within a constant mean rate. Examples include counting the number of cars passing a point in an hour or the number of emails received in a day, given a constant average rate at which these events occur.
- The main parameter of the Poisson distribution is \(\lambda = np\), which is the average number of occurrences in the given time interval.
- The formula for the Poisson probability is: \[P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}\], where \(e\) is the base of the natural logarithm, and \(!k\) is the factorial of 'k'.
This distribution is typically used when calculating the probability of a certain number of events happening within a constant mean rate. Examples include counting the number of cars passing a point in an hour or the number of emails received in a day, given a constant average rate at which these events occur.
Approximation Error
Approximation error is the difference between the real data or exact computation and an approximation that uses a simplified alternative like the Poisson distribution. Understanding and managing this error is essential in the field of statistics, as it guides how close an approximation is to reality.
In practice, one might compare the calculated values from both the binomial and Poisson distributions to comprehend how significant the approximation error is, as seen in the example exercises. Always remember to evaluate if the simplification justifies the inaccuracy.
- Approximation errors matter when choosing whether to approximate a binomial distribution using a Poisson distribution.
- These errors are usually smaller when the number of trials 'n' is large and the probability of success 'p' is small, making the events rare.
- The magnitude of the error can influence decision-making in various analytical approaches and processes including data modeling and predictive analytics.
In practice, one might compare the calculated values from both the binomial and Poisson distributions to comprehend how significant the approximation error is, as seen in the example exercises. Always remember to evaluate if the simplification justifies the inaccuracy.
