Question

If \(X\) is a Poisson variate such that \(P(X=2)=P(X=3)\), then \(P(X=0)=\ldots\)

Short answer

\(P(X=0) = e^{-3}\).

Step-by-step solution

Understanding the Poisson Distribution

The Poisson distribution is used for counting the number of events that happen independently in a fixed interval of time or space. The probability of observing exactly \(k\) events in a Poisson distribution with mean \(\lambda\) is given by the formula: \(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\).

Setting the Given Condition

We are given that \(P(X=2) = P(X=3)\). Using the Poisson probability formula, this can be rewritten as: \(\frac{\lambda^2 e^{-\lambda}}{2!} = \frac{\lambda^3 e^{-\lambda}}{3!}\).

Simplifying the Equation

Simplify the equation from Step 2: \(\frac{\lambda^2}{2} = \frac{\lambda^3}{6}\). By canceling common terms, we get \(3 (\lambda^2) = \lambda^3\).

Solving for \(\lambda\)

Solve the equation \(3\lambda^2 = \lambda^3\) by factoring out \(\lambda^2\) to get \(\lambda(\lambda - 3) = 0\). This yields solutions \(\lambda = 0\) or \(\lambda = 3\). Since \(\lambda\) must be positive, we choose \(\lambda = 3\).

Calculating \(P(X=0)\)

Using \(\lambda = 3\), calculate \(P(X=0)\) using the Poisson formula: \(P(X=0) = \frac{3^0 e^{-3}}{0!} = e^{-3}\).

Writing the Final Answer

Thus, \(P(X=0) = e^{-3}\).

Key concepts

Probability

Probability is the study of how likely an event is to occur. It applies to any situation involving chance or risk. A probability value ranges from 0 to 1, where 0 means the event will not happen and 1 indicates certainty that the event will happen. In the case of a Poisson distribution, which deals with the probability of a given number of events occurring in a fixed interval of time or space, probability is computed using a specific formula. This formula considers both the average number of events and the actual number of events observed in the interval. Understanding probability helps you predict outcomes and understand the likelihood of various scenarios.

Mean of Poisson Distribution

The mean of a Poisson distribution, denoted as \(\lambda\), is a crucial component in determining probabilities. This parameter represents the average number of times an event occurs within a specified period or space.

• In any Poisson distribution, the mean \(\lambda\) also equals the variance, which reflects how much variation or spread exists around the expected value.
• Choosing the correct \(\lambda\) is essential for accurate modeling. The better the mean is estimated, the more reliable the results will be.

Understanding the mean of the Poisson distribution provides a foundation for solving related probability calculations, forming the base of many real-world applications.

Poisson Distribution Formula

The Poisson distribution formula is used extensively in calculating probabilities for events in fixed intervals. The formula is given by: \[ P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \] In this formula:

• \(P(X=k)\) represents the probability of observing exactly \(k\) events.
• \(\lambda\) is the mean number of events in the interval.
• \(e\) is a mathematical constant approximately equal to 2.71828.
• \(k!\) denotes the factorial of \(k\), which is the product of all positive integers up to \(k\).

Using this formula allows you to calculate the likelihood of a particular number of occurrences, offering insights and precise predictions for event modeling.

Solving Equations with Poisson Distribution

In solving equations with Poisson distribution, it's important to apply the distribution formula to set up your equations properly. Consider a situation where you know two probabilities are equal, like \(P(X=2) = P(X=3)\). To solve this:1. **Write the equation using the Poisson formula**: - For \(P(X=2)\): \( \frac{\lambda^2 e^{-\lambda}}{2!} \) - For \(P(X=3)\): \( \frac{\lambda^3 e^{-\lambda}}{3!} \)2. **Simplify the equation**: Cancel common terms to get a simpler equation that can be solved for \(\lambda\). In the example above, simplifying results in \(3\lambda^2 = \lambda^3\).3. **Solve for \(\lambda\)**: Factor and solve the simplified equation. This yields \(\lambda = 0\) or \(\lambda = 3\). Since \(\lambda\) must be positive, choose \(\lambda = 3\).4. **Use the value of \(\lambda\)** to find other probabilities, such as \(P(X=0)\), which with \(\lambda = 3\) is \(e^{-3}\).Each step involves careful manipulation of numbers and understanding the behavior of the distribution to achieve the desired solution.