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