Respuesta :
Using the normal approximation to the Poisson distribution, it is found that there is a 0.460 = 46% probability that 10 squared centimeters of dust contains more than 10010 particles.
What is the Poisson distribution?
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by:
[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]
The parameters are:
- x is the number of successes
- e = 2.71828 is the Euler number.
- [tex]\mu[/tex] is the mean in the given interval.
Normal Probability Distribution
The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The z-score measures how many standard deviations the measure is above or below the mean.
- Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.
- The Poisson distribution can be approximated to the normal distribution with the same mean and standard deviation [tex]\sigma = \sqrt{\mu}[/tex].
In this problem, we have that the mean and the standard deviation are given as follows:
[tex]\mu = 10 \times 1000 = 10000, \sigma = \sqrt{10000} = 100[/tex]
The probability that it contains more than 10010 particles is one subtracted by the p-value of Z when X = 10010, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{10010 - 10000}{100}[/tex]
Z = 0.1
Z = 0.1 has a p-value of 0.5398.
1 - 0.5398 = 0.4602.
0.460 = 46% probability that 10 squared centimeters of dust contains more than 10010 particles.
More can be learned about the normal distribution at https://brainly.com/question/24537145
#SPJ1
