Problem 81 - Calculus for Biology and Medicine

Question

Suppose \(X\) is Poisson distributed with parameter \(\lambda=1.5\). Find the probability that \(X\) exceeds \(3 .\)

Step-by-Step Solution

Answer

The probability that \(X\) exceeds 3 is approximately 0.0655.

Step 1: Understanding the Poisson Distribution

The Poisson distribution is used to model the probability of a given number of events happening in a fixed interval of time or space. The parameter \(\lambda\) represents the average number of occurrences in that interval.

Step 2: Probability of a Poisson Random Variable

The probability that a Poisson random variable \(X\) takes on a value of \(k\) is given by the formula: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \] where \(k\) is a non-negative integer, and \(e\) is approximately 2.71828.

Step 3: Interpreting 'Exceeds 3'

To find the probability that \(X\) exceeds 3, we interpret \(X > 3\) which is equivalent to calculating the cumulative probability \( P(X \leq 3) \) and subtracting it from 1.

Step 4: Calculating \(P(X \leq 3)\)

Calculate \(P(X \leq 3)\) using the cumulative Poisson probability: \[ P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \]. Each is calculated using the Poisson formula with \(\lambda = 1.5\).

Step 5: Calculating Individual Probabilities

- \( P(X = 0) = \frac{e^{-1.5} \cdot 1.5^0}{0!} = e^{-1.5} \)- \( P(X = 1) = \frac{e^{-1.5} \cdot 1.5^1}{1!} = 1.5e^{-1.5} \)- \( P(X = 2) = \frac{e^{-1.5} \cdot 1.5^2}{2!} = 1.125e^{-1.5} \)- \( P(X = 3) = \frac{e^{-1.5} \cdot 1.5^3}{3!} = 0.5625e^{-1.5} \)

Step 6: Finding Cumulative Probability \(P(X \leq 3)\)

Sum the probabilities from Step 5: \[ P(X \leq 3) = e^{-1.5} (1 + 1.5 + 1.125 + 0.5625) = e^{-1.5} \, \cdot \, 4.1875 \].

Step 7: Probability that \(X > 3\)

The probability that \(X > 3\) is \[ P(X > 3) = 1 - P(X \leq 3) = 1 - e^{-1.5} \, \cdot \, 4.1875 \].

Step 8: Numerical Calculation

Calculate \(e^{-1.5} \approx 0.22313\). Then, \(P(X \leq 3) \approx 0.22313 \, \cdot \, 4.1875 = 0.93449\). Therefore, \(P(X > 3) \approx 1 - 0.93449 = 0.06551\).

Key Concepts

Probability Calculation

Probability calculation within the context of the Poisson distribution is an essential concept to grasp. The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space. This is especially useful in scenarios where events are rare or random, such as the number of emails you receive in an hour, or the number of natural disasters in a year.

The core formula for calculating the probability of a Poisson random variable, say, achieving exactly a particular value \(k\) is \( P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \). Here, \( \lambda \) is the average rate at which the events occur, and \( e \approx 2.71828 \) is the base of natural logarithms. This formula applies only for non-negative integer values of \( k \).

• \( \lambda \) represents the expected number of events.
• \( k \) is the specific number of events for which we're finding the probability.
• The use of the exponential function \( e \,^{-\lambda} \) helps model the decreasing probability of observing extremely high counts when \( \lambda\) is small.

The application of this formula often involves simple but rigorous calculations, where the recognition of parameters and correct substitution into the formula are critical.

Cumulative Probability

Cumulative probability in the Poisson context is all about understanding the probability of a variable being less than or equal to a certain value. For instance, it’s not just about achieving exactly three occurrences but any count up to and including three. When examining a problem stating "probability exceeds 3," the solution requires calculating the probability of reaching up to and including three events, then subtracting this from one to get the exceeding value.

For example, the cumulative probability \(P(X \leq 3)\) would mean that we sum up the probabilities of \(P(X = 0)\), \(P(X = 1)\), \(P(X = 2)\), and \(P(X = 3)\). Using the Poisson probability formula mentioned above, each of these can be calculated and summed:

• Calculate each probability separately using the formula.
• Sum these probabilities to get \(P(X \leq 3)\).
• Subtract this cumulative probability from 1 to find the probability of exceeding 3: \( P(X > 3) = 1 - P(X \leq 3) \).

This method helps simplify calculating probabilities for cases where the random variable needs to exceed a certain point.

Random Variable

A random variable in probability and statistics is a variable whose possible values are numerical outcomes of a random phenomenon. The term itself highlights uncertainty—there's a 'random' aspect involved, which is where probability theory plays a crucial role.

In the Poisson distribution, the random variable \(X\) often signifies counts of events. For instance, if \(X\) follows a Poisson distribution with parameter \(\lambda=1.5\), \(X\) could represent the random number of times an event occurs within a given timeframe.

• The random variable \(X\) can take any non-negative integer values (0, 1, 2,...).
• Each possible value of \(X\) is associated with a specific probability, calculated using the Poisson formula.
• The role of \(X\) highlights an essential statistical tool for modeling event counts where outcomes are not deterministic.

Understanding random variables is fundamental for grasping broader statistical concepts, as they form the backbone of statistical inference and predictive modeling.