For a Poisson distribution with
:
a.

b.
from formula and table.
c.

d. Probability
exceeds its mean by more than one standard deviation:

For a Poisson distribution with a mean
:
a. To find
using Table A.2 (Poisson cumulative distribution table), locate the row for
and find the corresponding value for
. The value in the table represents
, which is approximately 0.999.
b. To determine
using the probability mass function (pmf) formula for the Poisson distribution:
![\[P(X = k) = \frac{{e^(-\mu) \cdot \mu^k}}{{k!}}\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/ydzffnh1sx1y8efsai490zfaax6q6kfhng.png)
For
and
:
![\[P(X = 2) = \frac{{e^(-1) \cdot 1^2}}{{2!}} = \frac{{e^(-1)}}{{2}} \approx 0.1839\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/6wrw3pmn3sqanhblz3323ggtro4ymhmait.png)
You can also verify this value using Table A.2, where the row for
and column for
gives approximately

c. To find
, you can sum the individual probabilities
,
, and
using the Poisson pmf formula:
![\[P(X = 3) = \frac{{e^(-1) \cdot 1^3}}{{3!}} = \frac{{e^(-1)}}{{6}} \approx 0.0613\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/pigrgu0qnym072urd6fctkti2b1ejfdj74.png)
![\[P(X = 4) = \frac{{e^(-1) \cdot 1^4}}{{4!}} = \frac{{e^(-1)}}{{24}} \approx 0.0153\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/c2zemx6lz9dhy2sl5hrgg5jy4zxbo9juw3.png)
Therefore,

d. The standard deviation of a Poisson distribution is
. For
, the standard deviation
is
.
The probability that \(X\) exceeds its mean value by more than one standard deviation is the probability of
. This is the same as finding
.
![\[P(X > 2) = 1 - P(X \leq 2)\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/yaer28qdwmmfmnaeylcww0lvpfvshlp8ku.png)
Using Table A.2 for
and
,

So

complete the question
The article "Expectation Analysis of the Probability of Failure for Water Supply Pipes" (J. of Pipeline Systems Engr. and Practice, May 2012: 36-46) pro- posed using the Poisson distribution to model the num- ber of failures in pipelines of various types. Suppose that for cast-iron pipe of a particular length, the expected number of failures is 1 (very close to one of the cases considered in the article). Then X, the number of failures, has a Poisson distribution with μ = 1.
a. Obtain P(X ≤5) by using Appendix.
b. Determine P(X=2) first from the pmf formula and then from Appendix Table A.2.
c. Determine P(2 ≤ X ≤ 4).
d. What is the probability that X exceeds its mean value by more than one standard deviation?