give x has a poisson distribution with mean=1.6 what is the probability that x=9
To find the probability that x=9 given a Poisson distribution with mean = 1.6, we can use the formula for the probability mass function (PMF) of a Poisson distribution.
The PMF of a Poisson distribution is given by the formula P(x;λ) = (e^(-λ) * λ^x) / x!, where λ is the mean of the distribution and x is the specific value of interest.
In this case, λ = 1.6 and x = 9. Plugging these values into the formula, we get:
P(9;1.6) = (e^(-1.6) * 1.6^9) / 9!
Now, we need to calculate e^(-1.6) and 9!.
To evaluate e^(-1.6), we can use a calculator or computer software. The result is approximately 0.2019.
To calculate 9!, which represents the factorial of 9, we multiply all integers from 1 to 9 together: 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880.
Now we have all the necessary values to compute the probability:
P(9;1.6) = (0.2019 * 1.6^9) / 362,880
Evaluating this expression using a calculator or software, the approximated probability is:
P(x=9) ≈ 0.125