Questions: Define the Poisson probability function for a given mean: Consider a Poisson distribution with a mean of two occurrences per time period. (a) Write the appropriate Poisson probability function. p(x)=

Solution Steps

To define the Poisson probability function, we need to use the formula for the Poisson distribution, which is given by:

\[ p(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!} \]

where \( \lambda \) is the average rate (mean) of occurrences, \( x \) is the actual number of occurrences, and \( e \) is the base of the natural logarithm. In this case, the mean \( \lambda \) is 2.

The Poisson probability function is defined as:

\[ p(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!} \]

where \( \lambda \) is the mean number of occurrences, \( x \) is the number of occurrences, and \( e \) is the base of the natural logarithm.

For this problem, the mean \( \lambda = 2 \) and we want to find the probability of observing \( x = 3 \) occurrences. Substitute these values into the Poisson probability function:

\[ p(3; 2) = \frac{e^{-2} \cdot 2^3}{3!} \]

Calculate each component:

• \( e^{-2} \approx 0.1353 \)
• \( 2^3 = 8 \)
• \( 3! = 6 \)

Substitute these into the equation:

\[ p(3; 2) = \frac{0.1353 \cdot 8}{6} \approx 0.1804 \]

Final Answer