Respuesta :
Answer:
Variance = 5
Standard deviation = 2.236
0.2650257
Step-by-step explanation:
For a Poisson distribution is σ² = μ.
Given that:
Mean , μ of bankruptcies files per hour = 5
μ = 5
For a Poisson distribution :
P(x = x) = (μ^x * e^-μ) / x!
The Variance and standard deviation :
Variance : σ² = μ = 5
Standard deviation = sqrt(variance) = sqrt(5) = 2.236
B.) Find the probability that at most three businesses will file bankruptcy in any given hour.
P( x ≤ 3) = P(0) + P(1) + P(2) + P(3)
P(x = 0) = (5^0 * e^-5) / 0! = 0.0067379
P(x = 1) = (5^1 * e^-5) / 1! = 0.0336897
P(x = 2) = (5^2 * e^-5) / 2! = 0.0842243
P(x = 3) = (5^3 * e^-5) / 3! = 0.1403738
0.0067379 + 0.0336897 + 0.0842243 + 0.1403738
= 0.2650257
Using the Poisson distribution, it is found that:
a) The variance is of 5 while the standard deviation is of 2.24. It means that over many hours, the number of bankruptcies should be within 2.24 of 5.
b) 0.265 = 26.5% probability that at most three businesses will file bankruptcy in any given hour.
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, which is the same as the variance.
In this problem, the mean is of [tex]\mu = 5[/tex].
Item a:
- The variance is the same as the mean, of 5.
- The standard deviation is the square root of the variance, hence [tex]\sqrt{5} = 2.24[/tex]
The variance is of 5 while the standard deviation is of 2.24. It means that over many hours, the number of bankruptcies should be within 2.24 of 5.
Item b:
This probability is:
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)[/tex]
In which
[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-5}(5)^{0}}{(0)!} = 0.0067[/tex]
[tex]P(X = 1) = \frac{e^{-5}(5)^{1}}{(1)!} = 0.0337[/tex]
[tex]P(X = 2) = \frac{e^{-5}(5)^{2}}{(2)!} = 0.0842[/tex]
[tex]P(X = 3) = \frac{e^{-5}(5)^{3}}{(3)!} = 0.1404[/tex]
Then:
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.0067 + 0.0337 + 0.0842 + 0.1404 = 0.265[/tex]
0.265 = 26.5% probability that at most three businesses will file bankruptcy in any given hour.
A similar problem is given at https://brainly.com/question/16912674
