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Solve for P(0) Using the Poisson Probability Formula with μ=5
Mathematics (Statistics)
University (Introductory Statistics)
Question Content
Let $P(x)=\\frac{\\mu^{x}\\cdot e^{-\\mu}}{x!}$ and let $\\mu = 5$. Find $P(0)$. Round to six decimal places as needed.
Correct Answer
0.006738
Detailed Solution Steps
1
Step 1: Identify the given values and the target input. We know $\\mu=5$, and we need to calculate $P(x)$ when $x=0$.
2
Step 2: Recall the rules for factorials and exponents: $0! = 1$ (by definition) and any non-zero number raised to the power of 0 is 1, so $5^0=1$.
3
Step 3: Substitute $x=0$ and $\\mu=5$ into the formula: $P(0)=\\frac{5^{0}\\cdot e^{-5}}{0!}$.
4
Step 4: Simplify the expression using the rules from Step 2: $P(0)=\\frac{1\\cdot e^{-5}}{1}=e^{-5}$.
5
Step 5: Calculate the numerical value of $e^{-5}$. $e^{-5}\\approx0.006737947$.
6
Step 6: Round the result to six decimal places: $0.006738$.
Knowledge Points Involved
1
Poisson Probability Formula
The formula $P(x)=\\frac{\\mu^{x}\\cdot e^{-\\mu}}{x!}$ is the Poisson probability formula, used to calculate the probability of observing $x$ events in a fixed interval, where $\\mu$ is the average number of expected events. It is applied in scenarios where events occur independently at a constant average rate, such as counting customer arrivals or machine failures.
2
Factorial of 0
By mathematical definition, $0! = 1$. This rule is established to maintain consistency in combinatorial calculations and probability formulas (like the Poisson and binomial formulas) when the input for the factorial is 0.
3
Zero Exponent Rule
The zero exponent rule states that any non-zero real number raised to the power of 0 equals 1, i.e., $a^0=1$ for $a\\neq0$. This simplifies calculations in algebraic and statistical formulas when the exponent term is 0.
4
Rounding Decimal Places
Rounding to a specified number of decimal places involves looking at the digit immediately after the target decimal place. If that digit is 5 or greater, the last kept digit is rounded up; otherwise, it stays the same. This is used to present numerical results with a consistent, specified precision.
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