Question

Suppose that the number of spam emails that Alex receives has a Poisson distribution with µ = 2.3 per day. What is the probability that the number of spam emails Alex receives in a day is within one standard deviation of the mean? Clearly state the random variable of interest using the context of the problem and what probability distribution it follows.

Answer #1

**Answer:**

Given data

Mean = = 2.3

Standard deviation =

= 1.52

= Mean - Standard deviation

= 2.3 - 1.52

= 0.78

= Mean + Standard deviation

= 2.3 + 1.52

= 3.82

= 0.699

Poisson Distribution

P( k events in interval )

where

- is the average number of events per interval.
- e is the number 2.71828 ....(Euler's number ) the base of the natural logarithms
- k takes values 0,1,2,.....
- k! = k * (k-1) * ( k - 2) * -------- * 2 * 1 is the factorial of k.

This equation is the probability mass function (PMF) for a Poission distribution.

x | P(X = -x) |

0 | 0.10026 |

1 | 0.2306 |

2 | 0.26518 |

3 | 0.20331 |

Hence

= 0.699

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