The document discusses the geometric distribution, a discrete probability distribution that models the number of Bernoulli trials needed to get one success. It defines the geometric distribution and gives its probability mass function. Some key properties and applications are discussed, including: the mean is 1/p, the variance is q/p^2, where q is 1-p. It is used in situations like modeling the probability of events occurring after repeated independent trials with a constant probability of success each trial. Examples given include analyzing success rates in sports and deciding when to stop research trials.

1.  Geometric Distribution
1Geometric Distribution

2. 1. Definition and PDF.
2. Examples.
3. Uses.
4. Properties
5. Necessary.
6. Applications.
7. Mean and Variance.
8. Moment Generating Function.
9. Skewness And Kurtosis.
Geometric Distribution 2

3.  The Geometric Distribution a discrete probability
distribution. It is the distribution of the number of
trials needed to get the first success in repeated
Bernoulli trials.
 Satisfies the following conditions:
1. A trial is repeated until a success occurs.
2. The repeated trials are independent of each other.
3. The probability of success p is constant for each
trial. x represents the number of the trial in which
the first success occurs.
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4.  Probability of the 1st success on the Nth trial,
given a probability, p, of success
Geometric Distribution 4
ppxNP x 1
)1()( 

1
1
)1(0
1
1)1(
)1()1(
1
)1(
)1(
1
1
1
































p
p
p
p
p
p
pp
p
p
pp
pp
x
x
x
x
To show P(N=x) is a proper pdf:

5.  You play a game of chance that you can either win or
lose until you lose. Your probability of losing is p=0.57.
What is the probability that it takes five games until you
lose?
Geometric Distribution 5
Solution:
• Geometric with p = 0.57 ,
q = (1-p)
= 0.43 ,
x = 5
P(x=5) =
= 0.57(0.43)5-1
= 0.0194871657
1
)1( 
 x
pp

6. 1. In sports, particularly in baseball, a geometric
distribution is useful in analyzing the
probability.
2. In cost-benefit analyses, such as a company
deciding whether to fund research trials.
3. In time management, the goal is to complete
a task before some set amount of time.
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7. 1. The mean or expected value of a distribution gives
useful information about what average one would
expect from a large number of repeated trials.
2. The median of a distribution is another measure of
central tendency, useful when the distribution
contains outliers (i.e. particularly large/small values)
that make the mean misleading.
3. The mode of a distribution is the value that has the
highest probability of occurring.
4. The variance of a distribution measures how "spread
out" the data is. Related is the standard deviation--
the square root of the variance--useful due to being
in the same units as the data.
Geometric Distribution 7

8. 1. Geometric distribution used to model
probability. Which is very important in statistics
and in everyday life.
2. Businesses, governments, and families use to
make important decisions.
3. Geometric distributions provide statistical
models that show the possible outcomes of a
particular event. These models give people the
ability to make decisions.
Geometric Distribution 8

9. 1. used in Morkov chain models, particularly
meteorological mode of weather cycles and
precipitation amounts.
2. referred to as the failure time distribution.
3. used to describe the number of interviews that
have to be conducted by a selection board to
appoint the first acceptable candidate.
Geometric Distribution 9

10.  The Mean of the Geometric distribution is:
E(x) = 𝑥=1
∞
𝑥pqx-1
= p + 2pq + 3pq2 + 4pq3 + …………..
= p(1 + 2q + 3q2+ 4q3 + ………)
= p(1-q)-2
=
1
𝑝
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11.  The Variance of the Geometric distribution is:
E(X2)= 𝑥=1
∞
𝑥2
pqx-1
= p + 4pq + 9pq2 + 16pq3 +…………….
= p(1 + 4q + 9q2 + 16q3 +…………….)
= p(1 + 3q + 6q2 + 10q3 +…..) + pq(1 + 3q + 6q2 +……..)
= p (1-q)-3 + pq(1-q)-3
=
1
𝑝2 +
𝑞
𝑝2
Hence,
Variance= µ2 = E(x2) - [E(x)]2
=
1
𝑝2 +
𝑞
𝑝2 -
1
𝑝2
=
𝑞
𝑝2
Geometric Distribution 11

12.  Moment Generating Function of Geometric Distribution is
given bellow:
MX(t) = E(etx)
= 0
∞
𝑒 𝑡𝑥 𝑓 𝑥; 𝑝
= 0
∞
𝑒 𝑡𝑥
pqx
= p 0
∞
(𝑞𝑒 𝑡
) 𝑥
= p + pqet + pq2et2 + pq3et3 +…………
= p[1+ qet + (qet)2 + (qet)3+ ………..]
= p(1-qet)-1
=
𝑝
1−𝑞𝑒 𝑡
Geometric Distribution 12

13.  The moments of Geometric Distribution are
given bellow:
1. µ’1 =
𝑞
𝑝
2. µ’2 =
𝑞(𝑞+ 1)
𝑝2
3. µ’3 =
𝑞(𝑞2+4𝑞+1)
𝑝3
4. µ’4 =
𝑞(𝑞3+11𝑞2+11𝑞+1)
𝑝4
Geometric Distribution 13

14.  The 2nd, 3rd and 4th central moments of
Geometric Distribution are given bellow:
1. µ2 = µ’2 – (µ’1)2
=
𝑞
𝑝2
1. µ3 = µ’3 - 3µ’2µ’1 + 2(µ’1)3
=
𝑞(1+𝑞)
𝑝3
1. µ4 = µ’4 - 4µ’3µ’1 + 6µ’2(µ’1)2 - 3(µ’1)4
=
𝑞(𝑞2+7𝑞+1)
𝑝4
Geometric Distribution 14

15. The Skewness and Kurtosis of Geometric
Distribution are given bellow:
Skewness = 𝛽1 =
(µ3)2
(µ2)3 =
(1+𝑞)2
𝑞
Kurtosis = 𝛽2 =
µ4
(µ2)2 =
𝑞2+7𝑞+1
𝑞
Geometric Distribution 15

16. Have a good day
Geometric Distribution 16