Let $P(x)=\frac{\mu^{x} \cdot e^{-\mu}}{x !}$ and let $\mu=10$. Find $P(7)$
$P(7)=\square($ Round to four decimal places as needed.)
Answer
So, the final answer is \(\boxed{0.0901}\).
Step 1 ：Let's denote the function as \(P(x)=\frac{\mu^{x} \cdot e^{-\mu}}{x !}\) where \(\mu=10\). We are asked to find the value of \(P(7)\).
Step 2 ：The function \(P(x)\) is a Poisson distribution function. The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. In this case, \(\mu\) is the mean rate of occurrence.
Step 3 ：To find \(P(7)\), we need to substitute \(x=7\) and \(\mu=10\) into the function and calculate the result.
Step 4 ：After substituting the values, we get \(P(7) = 0.09007922571921599\).
Step 5 ：Rounding to four decimal places, we get \(P(7) = 0.0901\).
Step 6 ：So, the final answer is \(\boxed{0.0901}\).
