Question

Probability Mass Functions, Random Variables

Find a table of the Binomial random variable (include a picture of the table in your submission) and obtain the probability that in 20 independent trials, each of which has probability of success equal to 0.1, the number of successes is less than or equal to 3.

Repeat the problem using instead of a Binomial a Poisson with suitable parameter lambda.

Answer 1

Hello,

Suppose we conduct an experiment where the outcome is either "success" or "failure" and where the probability of success is p.

The PDF of bimomial distribution as,

x=1,2,3,........20;

p=0.1

then PDF table as,

Binomial distribution (n=20, p=0.1)
	f(x)	F(x)	1 - F(x)
x	Pr[X = x]	Pr[X ≤ x]	Pr[X > x]
0	0.122	0.122	0.878
1	0.27	0.392	0.608
2	0.285	0.677	0.323
3	0.19	0.867	0.133
4	0.09	0.957	0.043
5	0.032	0.989	0.011
6	0.009	0.998	0.002
7	0.002	1	4E-04
8	4E-04	1	1E-04
9	1E-04	1	0
10	0	1	0
11	0	1	0
12	0	1	0
13	0	1	0
14	0	1	0
15	0	1	0
16	0	1	0
17	0	1	0
18	0	1	0
19	0	1	0
20	0	1	0

Poission distribution is used Binomal distribution when the p is very small.

PDF of poisson distribution as,

X=1,2,3,......20

lambda=0.1

The probability table as,

Number of occurrences	Exact Probability	Cumulative Probability
0	0.905	0.905
1	0.09	0.995
2	0.005	1
3	0	1
4	0	1
5	0	1
6	0	1
7	0	1
8	0	1
9	0	1
10	0	1
11	0	1
12	0	1
13	0	1
14	0	1
15	0	1
16	0	1
17	0	1
18	0	1
19	0	1
20	0	1

From both the the tables, giving the similar results

Thanks