Question

Poisson vs. Binomial Let \(x\) be a binomial random variable with \(n=20\) and \(p=.1\). a. Calculate \(P(x \leq 2)\) using Table 1 in Appendix I to obtain the exact binomial probability. b. Use the Poisson approximation to calculate $$ P(x \leq 2) $$ c. Compare the results of parts a and b. Is the approximation accurate?

Short answer

Question: Calculate and compare the probabilities of \(P(x \leq 2)\) using the Binomial distribution with parameters (\(n=20\), \(p=0.1\)) and the Poisson approximation. Answer: The probability using the Binomial distribution is \(P_{\text{Binomial}}(x \leq 2) \approx 0.1589\), while the probability using the Poisson approximation is \(P_{\text{Poisson}}(x \leq 2) \approx 0.6767\). The difference between the two probabilities is quite significant in this case, indicating that the Poisson approximation might not be accurate for this problem.

Step-by-step solution

Binomial distribution parameters

Let's recall that a random variable \(x\) follows a Binomial distribution with parameters (\(n\), \(p\)), denoted by \(B(n, p)\). In this case, \(n=20\) and \(p=0.1\).

Calculate \(P(x \leq 2)\) using the Binomial distribution

Using Table 1 in Appendix I (or a calculator with built-in functions), we determine the Binomial probability for \(P(x \leq 2)\) as follows: $$ P(x \leq 2) = P(x = 0) + P(x = 1) + P(x = 2) $$ Using the given table, we find: $$ P(x \leq 2) \approx 0.0061 + 0.0384 + 0.1144 = 0.1589 $$

Poisson distribution parameter

To use the Poisson approximation, we need the parameter \(\lambda = n \times p\). In this case, \(\lambda = 20\times0.1 = 2\).

Calculate \(P(x \leq 2)\) using the Poisson approximation

Using the Poisson distribution with the parameter \(\lambda = 2\), we calculate \(P(x \leq 2)\) as follows: $$ P(x \leq 2) = P(x = 0) + P(x = 1) + P(x = 2) $$ The Poisson probability for \(x=k\) is given by: $$ P(x=k) = \frac{e^{-\lambda}\lambda^k}{k!} $$ Substitute the values for \(k=0,1,2\) and \(\lambda=2\) to find the probabilities: $$ P(x \leq 2) = \frac{e^{-2}2^{0}}{0!} + \frac{e^{-2}2^{1}}{1!} + \frac{e^{-2}2^{2}}{2!} ≈ 0.1353 + 0.2707 + 0.2707 = 0.6767 $$

Compare the results

We have obtained the probabilities using the Binomial distribution and the Poisson approximation: $$ P_{\text{Binomial}}(x \leq 2) \approx 0.1589 \\ P_{\text{Poisson}}(x \leq 2) \approx 0.6767 $$ Comparing the results, we can see that in this specific case the difference is quite significant, meaning that the Poisson approximation might not be accurate for this problem as it overestimates the probability. However, it is important to note that the Poisson approximation is generally more accurate for larger \(n\) and smaller \(p\) values, while this problem has a relatively small \(n\) and a relatively large \(p\).

Key concepts

Binomial Distribution

The binomial distribution is a common probability distribution that is used to model the number of successes in a fixed number of independent experiments. Each experiment, or trial, results in a success with probability \( p \) and a failure with probability \( 1-p \). The distribution is characterized by two parameters: \( n \), the number of trials, and \( p \), the probability of success on each trial. For any outcome, the probability is given by \[ P(x = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( \binom{n}{k} \) represents the combination of choosing \( k \) successes from \( n \) trials.In the exercise, we are given \( n = 20 \) and \( p = 0.1 \), with the task to compute \( P(x \leq 2) \). This involves summing the probabilities of obtaining 0, 1, and 2 successes. Using the binomial probability formula, calculations provide a probability of approximately 0.1589 for this scenario.

Poisson Distribution

While the binomial distribution is convenient for situations where the number of trials is fixed, the Poisson distribution is often used for counting the number of times an event occurs in a fixed interval of time or space. This distribution is characterized by the parameter \( \lambda \), which is the average rate of occurrence during that interval. It is particularly useful when \( n \) is large and \( p \) is small, which makes calculation with the binomial distribution cumbersome.The probability of observing \( x \) events in an interval is given by the formula:\[ P(x = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. In the exercise, \( \lambda \) is calculated by multiplying \( n \) and \( p \), resulting in \( \lambda = 2 \). Calculating \( P(x \leq 2) \) using the Poisson formula results in a probability of approximately 0.6767.

Approximation Accuracy

Approximation accuracy deals with how closely a simpler model, such as the Poisson distribution, can estimate a more complex model like the binomial distribution. In practice, the Poisson approximation is often employed when \( n \) is large and \( p \) is small, making \( np \) a moderate number. This condition allows the Poisson model to approximate the properties of the binomial distribution with fewer computations.In this exercise, the comparison between binomial and Poisson probabilities, \( P_{\text{Binomial}}(x \leq 2) \approx 0.1589 \) and \( P_{\text{Poisson}}(x \leq 2) \approx 0.6767 \), indicates a notable difference. The Poisson approximation overestimates the probability significantly. This discrepancy arises because the conditions \( n = 20 \) and \( p = 0.1 \) do not ideally fit the criteria where the Poisson approximation is accurate. Therefore, it's crucial to recognize that while approximations can simplify calculations, understanding their limitations ensures more accurate interpretations. In general, for small \( n \) and relatively larger \( p \), the binomial distribution should be employed directly rather than its Poisson approximation.