Basic Math Associated with Poisson Process

[nicholasjalonso]

In the following GARP 2020 Practice Exam question #65, we use the Poisson process for defaults at 1 and 0.

65. An analyst on the fixed-income trading desk observed that the number of defaults per year in the bond portfolio follows a Poisson process. The average number of defaults is four per year. Assuming defaults are independent, what is the probability that there is at most one default next year?

A. 6.58%
B. 7.33%
C. 9.16%
D. 25.00%

Correct Answer: C




As a general (and maybe dumb math) question, how do we get 0 in the denominator as equal to 1 which you then multiple by e^-4. I always was taught in math, by definition, that a denominator of 0 is undefined and not possible.

Thanks - just tying up some loose ends before my exam Saturday! You've been so helpful David

 

[dtammerz]

In the following GARP 2020 Practice Exam question #65, we use the Poisson process for defaults at 1 and 0:
View attachment 2957

As a general (and maybe dumb math) question, how do we get 0 in the denominator as equal to 1 which you then multiple by e^-4. I always was taught in math, by definition, that a denominator of 0 is undefined and not possible.

Thanks - just tying up some loose ends before my exam Saturday! You've been so helpful David

@nicholasjalonso I was also confused when looking at the soltuion of this same practice question last week. I found that zero factorial (0!) is always equal to 1, so it's not the same as plain zero.

https://en.wikipedia.org/wiki/Factorial#Factorial_of_zero

 

[nicholasjalonso]

Ah, thank you @dtammerz. I must have missed that math class back in the day lol. Thank you for that tidbit that's super helpful to know!