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?
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)
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