Homework Soursea PX
(a) P(X<6)
Solution
A random variable X is distributed by the binomial distribution with =15 =.1. Find the following probabilities, first using the binomial distribution and then using the Poisson approximation to the binomial distribution. Compare your results.
Binomial Probabilities Table
n=15
P=0.10
X
P(X)
0
0.2059
1
0.3432
2
0.2669
3
0.1285
4
0.0428
5
0.0105
6
0.0019
7
0.0003
8
0.0000
9
0.0000
10
0.0000
11
0.0000
12
0.0000
13
0.0000
14
0.0000
15
0.0000
(a) P(X<6) = 0.9978
(b) P(4 ) =0.0556
(c) = 0.0127
(d) =1.000
Mean/Expected number of events of interest: 15*0.1=1.5
POISSON.DIST Probabilities Table
X
P(X)
0
0.2231
1
0.3347
2
0.2510
3
0.1255
4
0.0471
5
0.0141
6
0.0035
7
0.0008
8
0.0001
9
0.0000
10
0.0000
11
0.0000
12
0.0000
13
0.0000
14
0.0000
15
0.0000
16
0.0000
17
0.0000
18
0.0000
19
0.0000
20
0.0000
(a) P(X<6) = 0.9955
(b) P(4 ) =0.0656
(c) = 0.0186
(d) =1.000
There is little difference in Binomial and Poisson probabilities when X < 6 or X <=8.
The difference is high when X > = 5 .
| Binomial Probabilities Table | ||
| n=15 P=0.10 | X | P(X) |
| 0 | 0.2059 | |
| 1 | 0.3432 | |
| 2 | 0.2669 | |
| 3 | 0.1285 | |
| 4 | 0.0428 | |
| 5 | 0.0105 | |
| 6 | 0.0019 | |
| 7 | 0.0003 | |
| 8 | 0.0000 | |
| 9 | 0.0000 | |
| 10 | 0.0000 | |
| 11 | 0.0000 | |
| 12 | 0.0000 | |
| 13 | 0.0000 | |
| 14 | 0.0000 | |
| 15 | 0.0000 |
