Question

2. Suppose that on average, there are 15 companies making their initial public offering of stock (IPO) each month. Write down the corresponding Poisson formula for a)-c), then use R to get the final answers.

a) What is the probability of few than 3 IPOs in a month?

b) What is the probability of at least 15 IPOs in a month?

c) What is the probability of few than 30 IPOs in a two-month period?

Answer 1

Poisson formula

P(k)= e^(-l)*l^k/k!
P(k)= probability of observing k events in an interval
l= lambda= average number of events per interval

a
probability of fewer than 3 IPOs in a month= probability of 0,1,2 IPOs in a month

k	P(k)	Formula (2 approaches)
0	0.000031%	=EXP(-15)*(15^0)/FACT(0)
1	0.000459%	=POISSON.DIST(2,15,TRUE(cumulative))
2	0.003441%
Total	0.003931%

Hence, probability of fewer than 3 IPOs in a month= 0.003931%

b

probability of at least 15 IPOs in a month= 1- probability of 0-14 IPOs in a month

k	P(k)
0	0.000031%
1	0.000459%
2	0.003441%
3	0.017207%
4	0.064526%
5	0.193579%
6	0.483947%
7	1.037029%
8	1.944430%
9	3.240717%
10	4.861075%
11	6.628739%
12	8.285923%
13	9.560681%
14	10.243587%
total	46.565371%
1- probability	53.434629%

Hence, probability of at least 15 IPOs in a month= 53.4346%

c

probability of few than 30 IPOs in a two-month period= total probability of having (x,y) in 2 months period where, x= number of IPOs in 1st month and y= number of IPOs in 2nd month, such that (x+y)<30. This method can be very exhaustive since the term value (x+y) could have 29 possible values where x and y can vary to add up to these 29 values. Another simpler approximation is to double the average to event average to get the two-period average. Hence, average number of IPOs over 2 months periods would be 30
probability of few than 30 IPOs in a two-month period= sum of 0 to 29 events occurring in the 2-months period.

k	P(k)
0	0.00000000001%
1	0.00000000028%
2	0.00000000421%
3	0.00000004211%
4	0.00000031582%
5	0.00000189492%
6	0.00000947459%
7	0.00004060540%
8	0.00015227025%
9	0.00050756750%
10	0.00152270249%
11	0.00415282497%
12	0.01038206242%
13	0.02395860557%
14	0.05133986909%
15	0.10267973817%
16	0.19252450907%
17	0.33974913366%
18	0.56624855610%
19	0.89407666752%
20	1.34111500128%
21	1.91587857326%
22	2.61256169081%
23	3.40768916193%
24	4.25961145241%
25	5.11153374289%
26	5.89792354949%
27	6.55324838833%
28	7.02133755892%
29	7.26345264716%
Total	47.57169861063%

Alternatively, we can use excel function which cumulative probability of 29 events as - =POISSON.DIST(29,30,TRUE)