Question

given that x has a poisson distribution with μ = 9, what is the probability that x = 4?
p(4) ≈ \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 \( \mu = 9 \), \( x = 4 \), \( e \) is the base of the natural logarithm (approximately 2.71828).

Step2: Substitute Values into Formula

Substitute \( \mu = 9 \), \( x = 4 \) into the formula:
First, calculate \( e^{-9} \approx 0.0001234098 \), \( \mu^{x} = 9^{4} = 6561 \), and \( x! = 4! = 24 \).
Then, \( P(4; 9) = \frac{0.0001234098 \times 6561}{24} \).

Step3: Perform Calculations

Calculate the numerator: \( 0.0001234098 \times 6561 \approx 0.8095 \).
Then divide by 24: \( \frac{0.8095}{24} \approx 0.0337 \). (More precisely, using a calculator for \( \frac{e^{-9} \times 9^{4}}{4!} \):
\( e^{-9} \approx 0.000123409804 \), \( 9^4 = 6561 \), \( 4! = 24 \)
So \( \frac{0.000123409804 \times 6561}{24} = \frac{0.8095014}{24} \approx 0.033729225 \))

Step4: Round to Four Decimal Places

Rounding 0.033729225 to four decimal places gives 0.0337.

Answer:

0.0337