Question

given that x has a poisson distribution with μ = 11, what is the probability that x = 7? p(7) ≈ \boxed{} (round to four decimal places as needed.)

Explanation:

Step1: Recall Poisson Probability Formula

The formula for the Poisson probability is \( P(x; \mu) = \frac{e^{-\mu} \mu^{x}}{x!} \), where \( \mu \) is the mean, \( x \) is the number of occurrences, \( e \) is the base of the natural logarithm (approximately 2.71828).

Step2: Identify Values

Here, \( \mu = 11 \) and \( x = 7 \).

Step3: Substitute Values into Formula

First, calculate \( e^{-\mu} \): \( e^{-11} \approx 1.67017 \times 10^{-5} \).
Then, calculate \( \mu^{x} \): \( 11^{7} = 19487171 \).
Next, calculate \( x! \): \( 7! = 5040 \).

Now, substitute these into the formula:
\( P(7; 11) = \frac{e^{-11} \times 11^{7}}{7!} = \frac{1.67017 \times 10^{-5} \times 19487171}{5040} \).

Step4: Perform Calculations

First, multiply the numerator: \( 1.67017 \times 10^{-5} \times 19487171 \approx 325.43 \).
Then, divide by the denominator: \( \frac{325.43}{5040} \approx 0.0646 \).

Answer:

\( P(7) \approx \boxed{0.0646} \)