Question

Find the probabilities for \(x\) using the Poisson formula. $$ \mu=2.5 ; P(x=0), P(x=1), P(x=2), \text { and } P(x \leq 2) $$

Short answer

Based on the given solution, calculate the probabilities for x=0, x=1, x=2, and x≤2 using the Poisson formula when the average number of occurrences (µ) is 2.5. The calculated probabilities are as follows: - P(x=0) ≈ 0.0821 - P(x=1) ≈ 0.2052 - P(x=2) ≈ 0.2565 - P(x≤2) ≈ 0.5438

Step-by-step solution

Calculate P(x=0)

To calculate the probability of x being 0, we have to plug 0 into the Poisson formula: $$P(x=0) = \frac{e^{-2.5} * 2.5^{0}}{0!}$$ Calculating the values, we get: $$P(x=0) = \frac{e^{-2.5} * 1}{1} = e^{-2.5} ≈ 0.0821$$

Calculate P(x=1)

To calculate the probability of x being 1, we have to plug 1 into the Poisson formula: $$P(x=1) = \frac{e^{-2.5} * 2.5^{1}}{1!}$$ Calculating the values, we get: $$P(x=1) = \frac{e^{-2.5} * 2.5}{1} ≈ 0.2052$$

Calculate P(x=2)

To calculate the probability of x being 2, we have to plug 2 into the Poisson formula: $$P(x=2) = \frac{e^{-2.5} * 2.5^{2}}{2!}$$ Calculating the values, we get: $$P(x=2) = \frac{e^{-2.5} * 6.25}{2} ≈ 0.2565$$

Calculate P(x ≤ 2)

To calculate the probability of x being less than or equal to 2, we need to find the sum of the probabilities for x being 0, 1, and 2: $$P(x \leq 2) = P(x=0) + P(x=1) + P(x=2)$$ Using the values we calculated in the previous steps: $$P(x \leq 2) = 0.0821 + 0.2052 + 0.2565 ≈ 0.5438$$ The probabilities for x using the Poisson formula are as follows: $$P(x=0) ≈ 0.0821, P(x=1) ≈ 0.2052, P(x=2) ≈ 0.2565, \text{ and } P(x \leq 2) ≈ 0.5438$$

Key concepts

Probability and Statistics

At its core, probability and statistics are branches of mathematics concerned with the analysis and interpretation of numerical data. Probability is used to predict the likelihood of an event occurring, such as the toss of a coin or the roll of a dice. In statistics, we collect, analyze, and present data which require understanding variances, averages, and patterns within the numbers.

When dealing with real-world problems, we often use these concepts to make informed decisions based on the outcomes' probabilities. For example, statisticians might use probability to estimate the likelihood of certain health outcomes based on medical data, or quality control analysts might use statistical tools to predict the likelihood of product defects.

With the given exercise scenario, we are focusing on a specific probability distribution called the Poisson distribution, which describes how often we can expect to see a certain number of events in a fixed interval of time or space.

Poisson Formula

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided these events occur with a known constant mean rate and independently of the time since the last event. The Poisson formula is expressed as:
\[\begin{equation}P(x=k) = \frac{e^{-\mu} * \$o\mu^k }{k!}\end{equation}\]
where:

• denotes the number of events,
• equals approximately 2.71828,
• \represents the mean number of events,
• is the factorial of .

It is particularly useful for calculating the probabilities of rare events over a large number of trials, such as the number of times a webpage might be accessed in an hour, or the number of calls a call center receives in a day. Implementing this formula requires a basic understanding of exponentiation, factorial operations, and the natural exponent .

Calculating Probabilities

Calculating probabilities involves determining the likelihood of various outcomes. In a Poisson distribution, these calculations can provide insights into the probability of a specific number of events occurring. Like our exercise, we're asked to find probabilities for different values of x using the Poisson formula.

To improve the exercise's understanding:- Using concrete examples can help illustrate the calculated probabilities. For instance, if x represents the number of calls to a helpline, then P(x=1) represents the probability of receiving one call.- It's also helpful to explain why we use the factorial in the Poisson formula: it accounts for the different sequences in which events can occur.- Visual aids such as probability trees or graphs can also make the connection between the formula and real-world implications clearer.

Therefore, once the concept of the Poisson distribution is understood, calculating probabilities becomes a straightforward task of plugging values into the Poisson formula and solving. This not only provides a single probability but can be built upon to find cumulative probabilities for a range of outcomes.