Questions: Let P(x) = (μ^x * e^(-μ)) / x! and let μ = 6. Find P(0). P(0) = (Round to six decimal places as needed.)

Transcript text: Let $P(x)=\frac{\mu^{x} \cdot e^{-\mu}}{x!}$ and let $\mu=6$. Find $P(0)$. $P(0)=$ $\square$ (Round to six decimal places as needed.)

Solution

Solution Steps

Step 1: Substitute Parameters

Given that $\mu = 6$ and $x = 0$, we substitute these values into the Poisson formula.

Step 2: Calculate Factorial

The factorial of $x$ is calculated as $x! = 1$.

Step 3: Compute Power and Exponential

We compute $\mu^x = 6^0 = 1$ and $e^{-\mu} = e^{-6} = 0.00248$.

Step 4: Perform Division

Dividing the product of $\mu^x$ and $e^{-\mu}$ by $x!$, we find $P(x) = \frac{\mu^x \cdot e^{-\mu}}{x!} = 0.00248$.

Step 5: Round as Needed

Rounding the result to 6 decimal places, we get $P(x) = 0.00248$.

Final Answer:

The probability of observing exactly $x = 0$ events, given a mean rate of $\mu = 6$, is $P(x) = 0.00248$.

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