Question

1)    75% of adult smokers started smoking before turning 18 years in a population. A random sample of 30 smokers 18 years or older are selected and the number of smokers who started smoking

before 18 is recorded.

1)   Find the probability that exactly 7 are smokers.

2)   Find the probability that at least 5 are smokers.

3)   Find the probability that fewer than 3 are smokers.

4)   Find the probability that between 4 and 7 of them, inclusive, are smokers.

5)   Find the mean and standard deviation of this binomial experiment.

6)   The mean value for an event X to occur is 2 in a day. Find the probability of event X to happen 3 times in a day.

      

8) Find the probability mass function of Poisson distribution.

Define moment generating function for discrete and continuous distribution.

9) Find the mean and variance of Poisson distribution using MGF.

Answer 1

1. 75% of adult smokers started smoking before turning 18 years in a population. A random sample of 30 smokers 18 years or older are selected and the number of smokers who started smoking before 18 is recorded.

Binomial distribution used, n=30, p=0.75

1.    Find the probability that exactly 7 are smokers.

P( x=7) =0.0000

1.    Find the probability that at least 5 are smokers.

P( x >=5) =1.000

1.    Find the probability that fewer than 3 are smokers.

P( x <3) =0.000

1.    Find the probability that between 4 and 7 of them, inclusive, are smokers.

P( 4<x<7) = P( x=4)+P( x=5)+P( x=6)+P( x=7)

1.    Find the mean and standard deviation of this binomial experiment.

Expectation = np = 22.5

Variance = np(1 - p) = 5.625

Standard deviation = 2.3717

6)The mean value for an event X to occur is 2 in a day. Find the probability of event X to happen 3 times in a day.

Poisson distribution used. λ=2 and x=3

P(X=x) = e^-l l^x / x!   

Mean =2

P( x=3) = 0.1804

Binomial Probabilities
Data
Sample size	30
Probability of an event of interest	0.75
Statistics
Mean	22.5
Variance	5.6250
Standard deviation	2.3717
Binomial Probabilities Table
	X	P(X)	P(<=X)	P(<X)	P(>X)	P(>=X)
	0	0.0000	0.0000	0.0000	1.0000	1.0000
	1	0.0000	0.0000	0.0000	1.0000	1.0000
	2	0.0000	0.0000	0.0000	1.0000	1.0000
	3	0.0000	0.0000	0.0000	1.0000	1.0000
	4	0.0000	0.0000	0.0000	1.0000	1.0000
	5	0.0000	0.0000	0.0000	1.0000	1.0000
	6	0.0000	0.0000	0.0000	1.0000	1.0000
	7	0.0000	0.0000	0.0000	1.0000	1.0000
	8	0.0000	0.0000	0.0000	1.0000	1.0000
	9	0.0000	0.0000	0.0000	1.0000	1.0000
	10	0.0000	0.0000	0.0000	1.0000	1.0000
	11	0.0000	0.0000	0.0000	1.0000	1.0000
	12	0.0000	0.0001	0.0000	0.9999	1.0000
	13	0.0002	0.0002	0.0001	0.9998	0.9999
	14	0.0006	0.0008	0.0002	0.9992	0.9998
	15	0.0019	0.0027	0.0008	0.9973	0.9992
	16	0.0054	0.0082	0.0027	0.9918	0.9973
	17	0.0134	0.0216	0.0082	0.9784	0.9918
	18	0.0291	0.0507	0.0216	0.9493	0.9784
	19	0.0551	0.1057	0.0507	0.8943	0.9493
	20	0.0909	0.1966	0.1057	0.8034	0.8943
	21	0.1298	0.3264	0.1966	0.6736	0.8034
	22	0.1593	0.4857	0.3264	0.5143	0.6736
	23	0.1662	0.6519	0.4857	0.3481	0.5143
	24	0.1455	0.7974	0.6519	0.2026	0.3481
	25	0.1047	0.9021	0.7974	0.0979	0.2026
	26	0.0604	0.9626	0.9021	0.0374	0.0979
	27	0.0269	0.9894	0.9626	0.0106	0.0374
	28	0.0086	0.9980	0.9894	0.0020	0.0106
	29	0.0018	0.9998	0.9980	0.0002	0.0020
	30	0.0002	1.0000	0.9998	0.0000	0.0002