The document discusses the binomial distribution, which models the number of successes in a fixed number of Bernoulli trials. Some key points: 1) A binomial experiment consists of n independent trials with two possible outcomes (success/failure) and a fixed probability p of success on each trial. 2) The binomial random variable X represents the number of successes in n trials. 3) The probability of getting exactly r successes in n trials is given by the binomial probability mass function P(X=r) = nCr * pr * q(n-r), where q = 1 - p. 4) For a binomial distribution, the mean is μ = np, the variance is σ2 = np

1. PROBABILITY DISTRIBUTIONS : BINOMIAL DISTRIBUTION
DEFINITIONS /CONCEPTS EXAMPLES
1 A random variable is a quantity that cannot be 1. Experiment:
predicted in advance but is determined by the A coin is tossed 4 times.
outcomes of an experiment.
2 A random variable which is countable is called a If getting a head is a success
discrete random variable. 1 1
then p = , q =
2 2
3 The number of success in an experiment is a
discrete random variable. The number of times the head appears is a
discrete random variable where
4 A Bernoulli trial is a trial which results in two X = {0, 1, 2, 3, 4, 5}
possible outcomes : success or failure
.
5 The probability of success is represented by p p q P(X = r)
and the probability of failure is represented by q. 0 4
Thus, p + q = 1. 4C  1   1 
0 4 0   
2 2
6 An experiment consisting of a fixed number of 1 3
Bernoulli trials is called a Binomial experiment. 4C  1   1 
1 3 1    
2 2
7 The discrete random variable X denotes the 2 2
number of successes in n repeated independent 4C  1   1 
2 2 2   
Bernoulli trials and is called Binomial random 2 2
variables. 3 1
4C  1   1 
8 Probability distribution of Binomial random 3 1 3    
variables is called Binomial Distribution. 2 2
4 0
4C  1   1 
9 Binomial distribution of X is expressed in the 4 0 4    
form of X ∼ B ( n, p ) where 2 2
r 0 1 2 3 4
n = the number of trials P(X = r) 0.0625 0.25 0.375 0.25 0.0625
p = the probability of success
P(X= r)
10 The probability of getting r successful trial out of
n trials is given by
P(X = r) = n C r p r q n− r
where
P = probability
X = binomial discrete random variable
r = number of successes (r = 0, 1, 2, …., n)
n = number of trials
p = probability of success ( 0 < p < 1 )
q = probability of failure (q = 1 - p ) r
0 1 2 3 4
11 For the Binomial Distribution: 2. Given that 40% of the students in a school wear
spectacles. From a sample of 10 students, calculate
mean = µ = np the mean, variance and standard deviation of the
number of students who wear spectacles.
var ians = σ 2 = npq p = 40% = 0.4, q = 1 – 0.4 = 0.6, n = 10
Mean, µ = np Variance, σ 2 = npq
s tan dard deviation = σ = npq
= 10 x 0.4
= 10x 0.4 x 0.6
=4
= 2.4
=
Standard deviation??
azadsbpisb 2008

2. azadsbpisb 2008