Question

given that x has a poisson distribution with μ = 1.6, what is the probability that x = 3?
p(3) ≈ \boxed{} (round to four decimal places as needed.)

Explanation:

Step1: Recall Poisson Probability Formula

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

Step2: Substitute Values

Here, \( \mu = 1.6 \), \( x = 3 \). Substitute into the formula:
\( P(3) = \frac{(1.6)^3 \times e^{-1.6}}{3!} \)

First, calculate \( (1.6)^3 = 1.6 \times 1.6 \times 1.6 = 4.096 \)
Then, calculate \( e^{-1.6} \approx e^{-1.6} \approx 0.2018965 \)
Next, calculate \( 3! = 3 \times 2 \times 1 = 6 \)

Step3: Compute the Probability

Now, plug these values into the formula:
\( P(3) = \frac{4.096 \times 0.2018965}{6} \)
First, multiply the numerator: \( 4.096 \times 0.2018965 \approx 0.8269 \)
Then, divide by 6: \( \frac{0.8269}{6} \approx 0.1378 \)

Answer:

\( 0.1378 \)