Page 191 - DMTH404_STATISTICS

Unit 14: Theoretical Probability Distributions

Let the event that an item is found to be defective be termed as a success. Thus, we are given n = Notes
12 and p = 0.1.
a 12 0 12
(i) P r = 0f = C 0.1 a f 0.9 a f = 0.2824
0
(ii) a 12 1 1 11
P r =1f = C 0.1 a f 0.9 a f = 0.3766
a 12 2 10
(iii) P r = 2f = C 0.1 a f 0.9 a f = 0.2301
2
(iv) P (r  ) 2 = P(r = 0) + P(r = 1) +P(r = 2)
= 0.2824 + 0.3766 + 0.2301 = 0.8891
(v) P(r  2) = 1 - P(0) - P(1) = 1 - 0.2824 - 0.3766 = 0.3410

Example 7: In a large group of students 80% have a recommended statistics book. Three
students are selected at random. Find the probability distribution of the number of students
having the book. Also compute the mean and variance of the distribution.
Solution.

Let the event that 'a student selected at random has the book' be termed as a success. Since the
group of students is large, 3 trials, i.e., the selection of 3 students, can be regarded as independent
with probability of a success p = 0.8. Thus, the conditions of the given experiment satisfies the
conditions of binomial distribution.
The probability mass function P r a f = C 0.8 a f 0.2 a f 3- r
r
3
r ,
where r = 0, 1, 2 and 3
The mean is np = 3 ´ 0.8 = 2.4 and Variance is npq = 2.4 ´ 0.2 = 0.48

Example 8:
(a) The mean and variance of a discrete random variable X are 6 and 2 respectively. Assuming
X to be a binomial variate, find P(5  X  7).

(b) In a binomial distribution consisting of 5 independent trials, the probability of 1 and 2
successes are 0.4096 and 0.2048 respectively. Calculate the mean, variance and mode of the
distribution.
Solution.
(a) It is given that np = 6 and npq = 2

npq 2 1 1 2 3
1
6
 q = = = so that p =  = and n = ´ = 9
np 6 3 3 3 2
Now P(5  X  7) = P(X = 5) + P(X = 6) +P(X = 7)

5 1 4 6 1 3 7 1 2
æ
2 ö æ ö
æ
2ö æ ö
2ö æ ö
æ
= 9 C 5 ç ÷ ç ÷ + 9 C 6 ç ÷ ç ÷ + 9 C 7 ç ÷ ç ÷
è
è
3 ø è ø
3 ø è ø
3 ø è ø
è
3
3
3
2 5 9 9 9 2 5
= 9 ë é C + C ´ 2 + C ´ 4ù = 9 ´ 438
û
6
5
7
3 3
(b) Let p be the probability of a success. It is given that
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