Question

Under what conditions can the Poisson random variable be used to approximate a probability associated with the binomial random variable?

Short answer

Answer: The Poisson distribution can be used to approximate the binomial distribution when the number of trials (n) is large (typically, n ≥ 20) and the probability of success (p) is small (with np ≤ 10). In this case, the binomial distribution converges to the Poisson distribution with mean λ = np. The probability of observing a certain number of successes (x) can be calculated using the Poisson probability mass function: P(X = x) = (e^{-λ} * λ^x) / x! where x is a non-negative integer, e is Euler's number (approximately 2.71828), and λ is the mean of the Poisson distribution.

Step-by-step solution

Identifying the conditions for Poisson approximation

The Poisson distribution can approximate a binomial distribution when the number of trials (n) is large, and the probability of success (p) for each trial is small. A typical rule of thumb is that n should be greater than or equal to 20, and np should be less than or equal to 10.

Convergence of binomial distribution to Poisson distribution

When the conditions mentioned in step 1 are met, we can use the Poisson distribution as an approximation for the binomial distribution. In this case, λ = np, where λ is the mean of the Poisson distribution.

Calculating probabilities using the Poisson distribution

Once the binomial distribution has been approximated by the Poisson distribution, the probability of observing a certain number of successes (x) can be calculated using the Poisson probability mass function: P(X = x) = (e^{-λ} * λ^x) / x! where x is a non-negative integer (0, 1, 2, ...), e is Euler's number (approximately 2.71828), and λ is the mean of the Poisson distribution.

Key concepts

Poisson random variable

Understanding the Poisson random variable is crucial when dealing with certain types of probability problems. A Poisson random variable is typically used to model the number of times an event happens in a fixed interval of time or space. Unlike events modeled with a binomial distribution, where we have a set number of trials, events described by a Poisson process can occur an unlimited number of times.

Imagine standing on a bridge and counting the number of cars passing in an hour, or a call center tracking the number of calls received in a day. These scenarios are ideal for the Poisson random variable as they involve events happening independently with a known average rate, and there is no natural upper limit for the number of occurrences.

Another key aspect is that the time or space between Poisson events follows an exponential distribution. When approximating binomial distribution with a Poisson random variable, we describe rare events in a large sample space. Moreover, the Poisson distribution is discrete, characterized by its mean, often denoted as \( \lambda \).

Binomial distribution

The binomial distribution models the number of successes in a fixed number of independent trials, given a consistent probability of success in each trial. In our educational content, it's important to emphasize the binary nature of this distribution: for each trial, there can only be two outcomes—usually termed a 'success' or a 'failure'.

For instance, consider flipping a fair coin 10 times, where 'success' could be defined as the coin landing on heads. Each flip is independent, and the probability of success (landing heads) remains constant at 0.5 throughout the trials. This scenario can be depicted using the binomial distribution. The essential factors that define it include the number of trials \( n \), the probability of success in each trial \( p \), and the number of successes we are interested in \( k \).

• The probability for exactly \( k \) successes among \( n \) trials is given by the formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
• The mean (expected number of successes) is \( np \).
• The variance (measure of variability around the mean) is \( np(1 - p) \).

With these basics, students can better understand the binomial framework and how it's applied in real-world situations.

Probability mass function

The probability mass function, often abbreviated as PMF, is a function that gives us the probability that a discrete random variable is exactly equal to some value. Essentially, it helps us create a complete distribution of probabilities for the possible outcomes of a discrete random variable, such as the number of heads in a series of coin tosses or the number of cars passing a checkpoint.

For the binomial distribution, the PMF is denoted as \( P(X = k) \) and directly gives the probability of getting exactly \( k \) successes in \( n \) independent trials. For the Poisson distribution, the PMF is particularly straightforward. It's important to approach this with the understanding that we're looking at the likelihood of a number of events occurring in a fixed period of space or time, assuming these events happen at a known constant mean rate and independently of the time since the last event:

• The PMF of a Poisson distribution with mean \( \lambda \) is:\[ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \]
• Here, \( \lambda \) is the average rate (mean) of successes,
• \( e \) is Euler's number (approximately 2.71828),
• \( x \) is the number of successes for which we're calculating the probability, and it must be a non-negative integer,
• The exclamation mark \( ! \) denotes a factorial, which is the product of all positive integers up to \( x \).

With these expressions, students are prepared to handle a range of problems involving discrete probability distributions.