Given that x has a Poisson distribution withµ= 2, what is the probability that x = 5 P(5) = (Round to four decimal places as needed.)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Problem Statement
**Given:**
x has a Poisson distribution with a mean (μ) of 2.
**Question:**
What is the probability that x = 5?
**Notation:**
\[ P(5) \approx \] ☐ (Round to four decimal places as needed.)
### Explanation
In this problem, you are asked to calculate the probability of a Poisson-distributed random variable being equal to 5, given a mean of 2. The Poisson distribution is commonly used to model the number of times an event occurs in a fixed interval of time or space.
The probability mass function (PMF) of a Poisson distribution is given by:
\[ P(x; \mu) = \frac{e^{-\mu} \mu^x}{x!} \]
Where:
- \( \mu \) is the average rate (mean),
- \( x \) is the number of occurrences (in this case, 5),
- \( e \) is approximately equal to 2.71828 (Euler's number),
- \( x! \) is the factorial of x.
### Calculation
To compute \( P(5) \):
1. Plug in \( \mu = 2 \) and \( x = 5 \) into the Poisson PMF formula.
2. Simplify and compute the result.
3. Round the result to four decimal places.
This exercise demonstrates the process of using the Poisson distribution for probability calculations, often applicable in fields such as statistics, biology, and telecommunications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9907483-a41f-491a-9684-8dee08aa60aa%2Ffb7400ab-4eb6-4ba4-815f-aa0a33d43e13%2Fdxm9ufa_processed.png&w=3840&q=75)
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