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CONTENT
1 History and about Discrete Probability Distribution
(Discrete + Cumulative + Binomial + Poisson)
2 Solve the given exercises questions
(Discrete, Cumulative, Binomial and Poisson)
3 3 Applications and 3 Uses of DISCRETE PROBABILITY
DISTRIBUTION
4 Conclusion for DISCRETE PROBABILITY DISTRIBUTION
(Discrete, Cumulative, Binomial, Poisson)
5 References
1
PAGE
2
3
9
11
12
I.
History and about Discrete Probability Distribution
Discrete Probability Distribution is a branch of probability theory that deals with
random variables that can only take on a finite or countably infinite number of
possible values. It is used to model random phenomena where the outcome is one of
a limited number of possible outcomes. Some examples of discrete random variables
include the number of heads obtained when flipping a coin or the number of cars
passing through an intersection in a given time period.
The concept of discrete probability distribution dates back to the 17th century, when
mathematicians such as Blaise Pascal, Pierre de Fermat, and Jacob Bernoulli began to
develop the theory of probability. In the 18th and 19th centuries, other
mathematicians such as Carl Friedrich Gauss, Adrien-Marie Legendre, and PierreSimon Laplace further developed the theory of probability, introducing new concepts
such as the normal distribution and the Poisson distribution.
The binomial distribution is a discrete probability distribution that describes the
number of successes in a fixed number of independent trials. It is named after the
Swiss mathematician Jacob Bernoulli, who introduced it in the early 18th century. The
binomial distribution has many practical applications, such as in quality control,
genetics, and polling.
The Poisson distribution is another important discrete probability distribution, which
describes the probability of a given number of events occurring in a fixed interval of
time or space, when these events occur randomly and independently of each other.
It is named after the French mathematician Siméon Denis Poisson, who introduced it
in the early 19th century. The Poisson distribution has many practical applications,
such as in the study of rare events, such as accidents, natural disasters, and rare
diseases.
The cumulative probability, also known as the cumulative distribution function (CDF),
is a function that describes the probability of a random variable taking a value less
than or equal to a given value. It is obtained by summing the probabilities of all the
outcomes that are less than or equal to the given value. The CDF is a fundamental
concept in probability theory and is used in many practical applications, such as in
finance, insurance, and engineering.
Discrete probability distributions are important in many fields, including finance,
economics, engineering, and physics. They can be used to model various phenomena
and to make predictions about future outcomes based on past data.
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II.
Solve the given exercises questions
(1)
Second box
1
3
5
First box
X
2
(1,2)
(3,2)
(5,2)
6
(1,6)
(3,6)
(5,6)
8
(1,8)
(3,8)
(5,8)
= the sum of the numbers on the two cards
= 3,5,7,9,11,13
(i)
X
X2
P(X)
X2P(X)
XP(X)
3
9
1
9
9
9
3
9
5
25
1
9
25
9
7
49
2
9
98
9
9
81
2
9
162
9
11
121
2
9
242
9
13
169
1
9
169
9
5
9
14
9
18
9
22
9
13
9
(ii)
Mean = E(X) = ∑ XP(X) = 8
E(X2) = ∑ 𝑋 2 P(X) = 78
Standard Deviation
1
3
1
3
= √𝑉𝑎𝑟𝑖𝑒𝑛𝑐𝑒
= √𝐸 (𝑋 2 ) − [𝐸 (𝑋)]2
2
235
25
=√ −( )
3
3
3
∑ P(X) = 1
∑ 𝑋 2 P(X) = 78
∑ XP(X) = 8
1
3
1
3
=√
235
=√
705−625
625
9
9
√80
3
=
=
3
−
4√5
3
(2)
F(X) =
𝑥2
9
for x = 1,2,3
1
4
9
9
9
9
F(1) = , F(2) = , F(3) =
Probability Distribution of X
X
F(X)
1
1
9
2
4
9
3
9
9
P(X)
1
9
3
9
5
9
P (x = 2)
= F(2) – F(1)
=
4
9
1
-
9
=
3
9
=
1
3
E(X) = ∑ 𝑋 𝑃(𝑋) = x1P1 + x2P2 + x3P3
1
3
5
9
9
9
= (1 × ) + (2 × ) + (3 × )
=
=
1
9
22
9
=2
E(3X – 3)
= 3E(X) – 3
=
=
22
3
−
9
3
13
3
4
+
4
9
6
9
+
15
9
=4
1
3
(3)
N=7
X
= no of days Mg Mg will be late for work
= 0, 1, …, 7
1
P =
10
,q=1–p=
9
10
(i)
P (he will never be late) = P (X = 0)
1 0
9 7
10
10
= 7C0 ( ) ( )
(ii)
P (he will be late on exactly three)
= P (X = 3)
1 3
9 4
10
10
= 7C3 ( ) ( )
(iii)
P(he will be late at most three days)
= P (X ≤ 3)
= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
1 0
9 7
1 1
9 6
1 2
9 5
10
10
10
10
10
10
1 3
9 4
10
10
= 7C0 ( ) ( ) + 7C1 ( ) ( ) + 7C2 ( ) ( ) +
7
C3 ( ) ( )
(4).
Unfair dice which 6 is 3 times of other
Let probability in each case of getting 1,2,3,4,5 = a
Probability of getting 6 = 3a
5
a + a + a + a + a + 3a = 1
8a = 1
a=
1
1
8
8
P (getting 1 or 2) = +
=
2
1
8
=
8
1
4
N = 10
X = no: of times of getting 1 or 2 = 0,1,2, … ,10
p = P (getting 1 or 2) =
1
4
q = P (not getting 1 or 2) = 1 – p =
3
4
P(X = x) = NCx 𝑝 𝑥 𝑞𝑁−𝑥 , X = 0, 1, … , N
(i)
P(at least two times of getting 1 or 2)
= P (X ≥ 2)
= 1 – P(X < 2)
= 1 – [P(X = 0) + P(X=1)]
= 1 – [10C0 𝑝0 𝑞10 + 10C1 𝑝1 𝑞9 ]
1 0 3 10
= 1 − [10C0 ( ) ( )
4
4
1 1 3 9
+ 10C1 ( ) ( ) ]
4
4
(ii)
P (at most two times of getting 1 or 2)
= P (X ≤ 2)
= P (X = 0) + P (X = 1) + P (X = 2)
= 10C0 𝑝0 𝑞10 + 10C1 𝑝1 𝑞9 + + 10C2 𝑝2 𝑞8
1 0 3 10
= 10C0 ( ) ( )
4
4
1 1 3 9
1 2 3 8
4
4
+ 10C1 ( ) ( ) + 10C2 ( ) ( )
4
(iii) P (none of these getting 1 or 2)
4
= P(X = 0)
= 10C0 𝑝0 𝑞10
6
1 0 3 10
= 10C0 ( ) ( )
4
4
(5).
𝑁 = 400
𝑝 = 0.001
𝜆 = 𝜇 = 𝑁𝑝 = 400 × 0.001 = 0.4
𝑃 (𝑋 = 𝑘 ) =
𝑒 −𝜆 𝜆𝑘
𝑘!
, 𝑘 = 0 ,1 ,2 , …
𝑃(𝑒𝑥𝑎𝑐𝑡𝑙𝑦 3) = 𝑃(𝑋 = 3) =
(i)
=
𝑒 −0.4 ×(0.4)3
3!
0.6703×(0.4)3
6
= 0.0071
(ii) 𝑃(𝑛𝑜𝑡 𝑚𝑜𝑟𝑒 𝑡ℎ𝑎𝑛 2 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠)
= P (X ≤ 2)
= P (X = 0) + P (X = 1) + P (X=2)
=
𝑒 −0.4 ×(0.4)0
0!
+
𝑒 −0.4 ×(0.4)1
1!
+
𝑒 −0.4 ×(0.4)2
2!
= 𝑒 −0.4 [1 + 04 + 0.08]
= 0.6703 × 1.48
= 0.992
(iii)
𝑃(𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 2 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠) = 𝑃(𝑋 ≥ 2)
= 1 − 𝑃(𝑋 < 2)
= 1 − [𝑃(𝑋 = 0) + 𝑃(𝑋 = 1)]
=1−[
𝑒 −0.4 ×(0.4)0
0!
+
𝑒 −0.4 ×(0.4)1
= 1 − [𝑒 −0.4 (1 + 0.4)]
= 1 − [0.6703 × 1.4]
= 0.0616
7
1!
]
(6).
Let 𝑋 = no. of people who become seriously ill each year eating a certain
poisonous plant
𝜆 = 2 = 𝑚𝑒𝑎𝑛
𝑒 −𝜆 𝜆𝑘
𝑃(𝑋 = 𝑘) =
, 𝑘 = 0,1,2, . ..
𝑘!
(i)
𝑃(𝑎𝑡𝑚𝑜𝑠𝑡 3 𝑠𝑢𝑐ℎ 𝑖𝑙𝑙𝑛𝑒𝑠𝑠𝑒𝑠)
= 𝑃(𝑋 ≤ 3)
= 𝑃(𝑋 = 0) + 𝑃(𝑋 = 1) + 𝑃(𝑋 = 2) + 𝑃(𝑋 = 3)
=
𝑒 −2 ×(2)0
0!
+
𝑒 −2 ×(2)1
1!
+
𝑒 −2 ×(2)2
2!
+
𝑒 −2 ×(2)3
3!
4
= 𝑒 −2 [1 + 2 + 2 + ]
3
= 0.1353 ×
(ii)
19
= 0.8569
3
𝑃(3 𝑜𝑟 𝑚𝑜𝑟𝑒 𝑠𝑢𝑐ℎ 𝑖𝑙𝑙𝑛𝑒𝑠𝑠)
= 𝑃(𝑋 ≥ 3)
= 1 − 𝑃(𝑋 < 3)
= 1 − [𝑃(𝑋 = 0) + 𝑃(𝑋 = 1) + 𝑃(𝑋 = 2)]
=1−[
𝑒 −2 ×(2)0
0!
+
𝑒 −2 ×(2)1
1!
+
𝑒 −2 ×(2)2
2!
]
= 1 − 𝑒 −2 [1 + 2 + 2]
= 1 − (0.1353 × 5) = 0.3235
(iii)
𝑃(𝑒𝑥𝑎𝑐𝑡𝑙𝑦 3 𝑠𝑢𝑐ℎ 𝑖𝑙𝑙𝑛𝑒𝑠𝑠) = 𝑃(𝑋 = 3)
=
𝑒 −2 (2)3
3!
4
= 0.1353 × = 0.1804
3
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III.
3 Applications and 3 Uses of DISCRETE PROBABILITY
DISTRIBUTION
Applications
1. Genetics: Discrete probability distributions can be used to model the
probability of gene inheritance and the occurrence of genetic disorders. For
example, the Punnett square, which is a common tool used in genetics, uses
the laws of probability to predict the possible genotypes and phenotypes of
offspring based on the genotypes of their parents.
2. Risk Analysis: Discrete probability distributions can be used in risk analysis
to model the probability of various outcomes and estimate the potential
losses or gains associated with them. For example, in finance, discrete
probability distributions can be used to model the probability of stock prices
going up or down, and estimate the potential returns or losses associated
with each scenario.
3. Quality Control: Discrete probability distributions can also be used in quality
control to model the probability of defects or errors occurring in a production
process, and to estimate the effectiveness of quality control measures. For
example, a manufacturer can use the binomial distribution to model the
probability of defects in a batch of products, and use this information to
improve their production process.
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Uses
1. Decision Making: Discrete probability distributions can be used to support
decision making in a wide range of contexts. By modeling the probability
distribution of a random variable, decision makers can assess the potential
outcomes of different decisions and estimate the associated risks and
rewards. For example, a company can use a binomial distribution to model
2. Prediction: Discrete probability distributions can be used to make
predictions about future events based on past data. For example, the Poisson
distribution can be used to predict the number of customer arrivals at a store
based on past data, or the number of accidents in a certain area based on
historical records.
3. Statistical Analysis: Discrete probability distributions are an essential tool for
statistical analysis. By analyzing data and modeling it using a suitable
probability distribution, it is possible to make statistical inferences and draw
conclusions about the population being studied. For example, the binomial
distribution can be used to test hypotheses about the success rate of a new
drug treatment, or the Poisson distribution can be used to analyze the
frequency of defects in a manufacturing process.
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IV.
Conclusion
In conclusion, a discrete probability distribution is a function that describes the
probability of occurrence of each possible outcome in a discrete sample space. This
type of distribution is characterized by a finite or countably infinite set of possible
outcomes, each with a non-negative probability.
Examples of discrete probability distributions include the binomial distribution, the
Poisson distribution, and the geometric distribution. Each distribution has its own
specific properties and can be used to model different types of scenarios.
The binomial distribution is commonly used to model the number of successes in a
fixed number of independent trials, while the Poisson distribution is often used to
model the number of rare events occurring over a given period of time. The
geometric distribution is useful for modeling the number of independent trials
needed to achieve the first success.
Understanding discrete probability distributions is essential for many fields,
including statistics, economics, and engineering. By using probability distributions,
we can make predictions about the likelihood of different outcomes, which can
help us make better decisions and design more effective systems.
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V.
References
➢ Learner’s guide of Science II (Mathematics) Foundation Year, Department
of Mathematics, University Of Medicine (1) Yangon
➢ Khan Academy: Discrete Random Variables and Probability Distributions.
Retrieved from https://www.khanacademy.org/math/statisticsprobability/random-variables-stats-library/discrete-and-continuousrandom-variables/v/discrete-random-variables
➢ Stat Trek: Discrete Probability Distribution. Retrieved from
https://stattrek.com/probability-distributions/discrete-probability.aspx
➢ University of California, Davis: Discrete Probability Distributions. Retrieved
from https://www.stat.ucdavis.edu/~kienitz/classes/sta103/discrete.pdf
➢ Math is Fun: Probability Distribution. Retrieved from
https://www.mathsisfun.com/data/probability-distribution.html
➢ Wolfram MathWorld: Discrete Probability Distribution. Retrieved from
https://mathworld.wolfram.com/DiscreteProbabilityDistribution.html
➢ Towards Data Science: Understanding Discrete Probability Distributions.
Retrieved from https://towardsdatascience.com/understanding-discreteprobability-distributions-75ab9a1c22b6
➢ Statistic How To: Discrete Probability Distribution. Retrieved from
https://www.statisticshowto.com/discrete-probability-distribution/
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