Discrete probability distributions

Discrete Probability Distributions
Bipul Kumar Sarker
Lecturer
BBA Professional
Habibullah Bahar University College
Chapter-03

Bipul Kumar Sarker, Lecturer (BBA Professional), HBUC
Probability Distribution:
A probability distribution describes the behavior of a random variable. Many times
the observations obtained from different random experiments have a general type of
behavior.
Therefore, the random variable(s) associated with these experiments have a general
type of probability distribution.
So it can be represented by a single formula. This type of single formula
about the behavior of a random variable is known as probability distribution.

Discrete Probability Distribution:
The probability distribution of a discrete random variable that arise from some
statistical experiments is known as discrete probability distribution.
The following are some of the commonly discussed discrete distribution:
i. Uniform distribution
ii. Bernoulli distribution
iii.Binomial distribution
iv.Negative binomial distribution
v. Poisson distribution
vi.Geometric distribution
vii.Hypergeometric distribution

Bernoulli Trial:
When in the trial of a random experiment each trial is independent and in each the
number of possible outcomes are dichotomous, then the trial are called Bernoulli trial.
In Bernoulli trial, the outcome are known as ‘Success’ and ‘Failure’ or having a certain
quality or absence of that quality.
Here, the probability of success is denoted by p and the probability of failure is denoted by
(1-P)= q. These probability remains constant from trial to trial.

Examples of Bernoulli’s Trails are:
1) Toss of a coin (head or tail)
2) Throw of a die (even or odd number)
3) Performance of a student in an examination (pass or fail)

Bernoulli Distribution:
If a random variable x can assume only two values, failure and a success with
probabilities 𝑃 𝑥 = 1 = 𝑝 and 𝑃 𝑥 = 0 = 1 − 𝑝.
Then the distribution of x might be termed as a Bernoulli distribution with parameter p.
The probability distribution of Bernoulli variate x can be written as,
𝑷 𝑿 = 𝒙; 𝒑 = 𝒑 𝒙
𝒒 𝟏−𝒙
; 𝒙 = 𝟎, 𝟏
= 𝟎 ; 𝑶𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆

Example:
When a coin is tossed either head or tail will appear. If we consider having a head as the
„Success‟ and „Failure‟ otherwise then this be termed as Bernoulli trial and its
distribution as Bernoulli distribution.

Binomial Trial:
A binomial trial is the extension of Bernoulli trial. In a binomial experiment, the
trial is repeated several times. A binomial experiment should have the following properties:
i. The experiment consists of n repeated trials
ii. The trials are independent of each other
iii.The probability of success (failure) remains constant from trial to trial.

3.1.2 Binomial Distribution:
A random variable X is said to follow binomial distribution, if its probability mass
function is given by
Here, the two independent constants n and p are known as the „parameters‟ of the
distribution.
The distribution is completely determined if n and p are known. x refers the number of
successes.
Where, n = No. of trials
p = Probability of success
q = Probability of failure

3.1.3 Condition for Binomial Distribution:
We get the Binomial distribution under the following experimental conditions.
i. The number of trials ‘ n’ is finite.
ii. The trials are independent of each other.
iii. The probability of success ‘ p’ is constant for each trial.
iv. Each trial must result in a success or a failure.
The problems relating to tossing of coins or throwing of dice or drawing cards from a pack
of cards with replacement lead to binomial probability distribution.

3.1.4 Characteristics of Binomial Distribution:
1. Binomial distribution is a discrete distribution in which the random variable X
(the number of success) assumes the values 0,1, 2, ….n, where n is finite.
2. Mean = np, variance = npq and
clearly each of the probabilities is non-negative and sum of
all probabilities is 1 ( p < 1 , q < 1 and p + q =1, q = 1- p ).

Comment on the following: “ The mean of a binomial distribution is 5 and its variance is 9”
The parameters of the binomial distribution are n and p
We have, mean = np = 5
Variance = npq = 9
Which is not admissible since q cannot exceed unity. Hence
the given statement is wrong.

Example 2:
Eight coins are tossed simultaneously. Find the probability of getting at least six heads.
Solution:
Given that,
No. of trials, n = 8
The probability of getting a head, p =
1
2
The probability of getting a head, q = (1-p) =
1
2
Let, x be denoted the number of getting a head

Therefore, the probability function,
𝑷 𝑿 = 𝒙; 𝒑 = 𝒏 𝑪 𝒙
𝒑 𝒙
𝒒(𝒏−𝒙)
; x = 0, 1, 2, 3, 4, …, 7, 8
𝑷 𝑿 ≥ 𝟔 = 𝑷 𝑿 = 𝟔 + 𝑷 𝑿 = 𝟕 + 𝑷 𝑿 = 𝟖
Probability of getting at least six heads is given by
𝑷 𝑿 ≥ 𝟔 = 𝟖 𝑪 𝟔
(
𝟏
𝟐
) 𝟔
(
𝟏
𝟐
) 𝟐
+ 𝟖 𝑪 𝟕
(
𝟏
𝟐
) 𝟕
(
𝟏
𝟐
) 𝟏
+ 𝟖 𝑪 𝟖
(
𝟏
𝟐
) 𝟖
(
𝟏
𝟐
) 𝟎
𝑷 𝑿 ≥ 𝟔 = (
𝟏
𝟐
) 𝟖 { 𝟖 𝑪 𝟔
+𝟖 𝑪 𝟕
+ 𝟖 𝑪 𝟖
} =
37
256

Example 3:
Ten coins are tossed simultaneously. Find the probability of getting
i. At least seven heads
ii. Exactly seven heads
iii. At most seven heads

Solution: Given that,
No. of trials, n = 8
The probability of getting a head, p =
1
2
The probability of getting a head, q = (1-p) =
1
2
Let, x be denoted the number of getting a head
Therefore, the probability function,
𝑷 𝑿 = 𝒙; 𝒑 = 𝒏 𝑪 𝒙
𝒑 𝒙 𝒒(𝒏−𝒙) ; x = 0, 1, 2, 3, 4, …, 9, 10

i. Probability of getting at least seven heads is given by
ii. Probability of getting exactly 7 heads

iii. Probability of getting at most 7 heads
𝑷 𝑿 ≤ 7 = 𝑷 𝑿 = 0 + 𝑷 𝑿 = 1 + ⋯ … … . +𝑷 𝑿 = 7

Discrete probability distributions

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