Statistics/ Random variables ناوی قوتابی: بھزاد بھرام صابر ناوی مامۆستای بابەت : دلڤین حسن عمر رێکەوتی پێشکەش کردن: ٢٤ / ٦ / ٢٠٢٠ ناوی کۆلێژ : زانکۆی پۆلیتەکنیکی ئەندازیاری هەولێر ناوی بەشی زانستی : ئەندازیاری شارستانی

1. ‫هەولێر‬ ‫ئەندازیاری‬ ‫پۆلیتەکنیکی‬ ‫زانکۆی‬ : ‫کۆلێژ‬ ‫ناوی‬
‫شارستانی‬ ‫ئەندازیاری‬ : ‫زانستی‬ ‫بەشی‬ ‫ناوی‬
Statistics/ Random variables: ‫بابەت‬ ‫ناوی‬
‫ﺻﺎﺑر‬ ‫ﺑﮭرام‬ ‫ﺑﮭزاد‬ :‫ﻗوﺗﺎﺑﯽ‬ ‫ﻧﺎوی‬
: ‫بابەت‬ ‫مامۆستای‬ ‫ناوی‬
‫عمر‬ ‫حسن‬ ‫دلڤین‬
‫کردن‬ ‫پێشکەش‬ ‫رێکەوتی‬
:
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/
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2. 2
Contents
Introduction:.......................................................................................................................................3
Calculate binomial distribution:.......................................................................................................4
Calculate poisson distribution: .........................................................................................................5
Calculate normal distribution:..........................................................................................................6
Normal Distribution Curve:..............................................................................................................7
Summary and Learning Outcomes: .................................................................................................8
Discussion: ........................................................................................................................................11
Reference: .........................................................................................................................................12

3. 3
Introduction:
A random variable is a numerical description of the outcome of a statistical experiment. A random
variable that may assume only a finite number or an infinite sequence of values is said to be
discrete; one that may assume any value in some interval on the real number line is said to be
continuous.
Binomial distribution: The binomial distribution is a discreet distribution displaying data the only
two outcomes and each trail includes replacement. Such as: Pass/Fail, Go/no-Go, Success/Failure,
In/Out, Hot/Cold, Male/Female, High/Low, Heads/Tails, Defective/ not Defective, Right-hand/
Left-hand.
Poisson distribution: The normal distribution can also be used to approximate the Poisson
distribution whenever the parameter lambda, the expected number of success, equals or exceeds 5.
Since the value of the mean and the variance of a passion distribution are the same.
Normal distribution: Among the special probability densities we shall study in this part, the normal
probability density, usually referred to simply as the normal distribution, is by far the most
important. They found that the patterns (distributions) they observed were closely approximated by
a continuous distribution which they referred to as the normal curve of errors. The equation of the
normal probability density is shown in below.

4. 4
Calculate binomial distribution:
40
0.65
.
40 0.65 26
. ( ) . (1 )
. ( ) 40 (1 0.65) 3.0166
40
0.65
0,1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
26,27,28,29,30,31,32,33,34,35,36,37,38,3
:
:
Solu
D
n
p
n p
S
a
D S n p p
S
n
at
x
ti
D S
on
p





  
  
   



 
 
0 40 0
9
9
.
,40
40!
0 65 (1 0.65 1
( ) . .(1 )
!
! !
) 5.790576106764 1
0! 40 0 !
(0)
x n x
n
P x p p
x
n n
x x n x
P E


 
 
 
 

 


 

 

 


P(0) = 5.7905761067641E-19
P(1) = 4.3015708221676E-17
P(2) = 1.5577831477421E-15
P(3) = 3.6644994046886E-14
P(4) = 6.2950864773401E-13
P(5) = 8.4174299182719E-12
P(6) = 9.1188824114613E-11
P(7) = 8.2256041344202E-10
P(8) = 6.3014003101183E-9
P(9) = 4.160924649221E-8
P(10) = 2.3955037623372E-7
P(11) = 1.2133071004046E-6
P(12) = 5.4454378196729E-6
P(13) = 2.1781751278691E-5
P(14) = 7.8014231620619E-5
P(15) = 0.00025113152655018
P(16) = 0.00072872987615009
P(17) = 0.001910619507217
P(18) = 0.0045339304179198
P(19) = 0.0097496548836471
P(20) = 0.019011827023112
P(21) = 0.03362636072115
P(22) = 0.053933188948858
P(23) = 0.078387367789023
P(24) = 0.10311671596056
P(25) = 0.12256158239884
P(26) = 0.13131598114161
P(27) = 0.12645242628451
P(28) = 0.10903295939838
P(29) = 0.083788875202698
P(30) = 0.057056234066599
P(31) = 0.034181154049115
P(32) = 0.017853549213154
P(33) = 0.0080379615505107
P(34) = 0.0030733382399012
P(35) = 0.00097845054168282
P(36) = 0.00025237811591025
P(37) = 5.0670509758043E-5
P(38) = 7.4291348893371E-6
P(39) = 7.0753665612734E-7
P(40) = 3.2849916177341E-8

5. 5
Calculate poisson distribution:
0 26
0,1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
26,27,28,29,3
3
0,31,32,3
)
26
. ( )
. ( ) 26 5.099
0 5.10908
3
90280
,34,35,36,37,38,39,40.
.
(
!
26 .
( )
!
:
:
0
63
x
D
S
at
x
S D S
S D S
P
e
e
x
oluti
x
a
o
P
n







 




 

12
E 
Mean of the Poisson Distribution, μ: 26
Standard Deviation of the Poisson Distribution, S: 5.0990195135928
P(0) = 5.1090890280633E-12
P(1) = 1.3283631472965E-10
P(2) = 1.7268720914854E-9
P(3) = 1.4966224792874E-8
P(4) = 9.7280461153678E-8
P(5) = 5.0585839799912E-7
P(6) = 2.1920530579962E-6
P(7) = 8.1419113582716E-6
P(8) = 2.6461211914383E-5
P(9) = 7.6443501085995E-5
P(10) = 0.00019875310282359
P(11) = 0.00046978006121939
P(12) = 0.0010178567993087
P(13) = 0.0020357135986173
P(14) = 0.0037806109688608
P(15) = 0.006553059012692
P(16) = 0.010648720895625
P(17) = 0.016286279016837
P(18) = 0.023524625246543
P(19) = 0.032191592442638
P(20) = 0.041849070175429
P(21) = 0.051813134502912
P(22) = 0.061233704412533
P(23) = 0.069220709335907
P(24) = 0.074989101780565
P(25) = 0.077988665851788
P(26) = 0.077988665851788
P(27) = 0.075100196746166
P(28) = 0.069735896978583
P(29) = 0.062521838670454
P(30) = 0.054185593514393
P(31) = 0.045445981657233
P(32) = 0.036924860096502
P(33) = 0.029092314015426
P(34) = 0.022247063658855
P(35) = 0.016526390146578
P(36) = 0.011935726216973
P(37) = 0.0083872670713864
P(38) = 0.0057386564172644
P(39) = 0.0038257709448429
P(40) = 0.0024867511141479

6. 6
Calculate normal distribution:
Note:We can use table but I use Modern scientific calculators
li
26
. or (S) =3.0166
1- (22 27).
2- ( 26).
3- ( 30) ( 30).
A.1 standard Normal Proba
e TI-84 .
PLU
bilities
1-
S
:
:
k
(22
Da
So
S D
p X
p X
p X o
lution
t
r X
p
a
 
 

 
 27)
2- ( 26) 0.5
3- ( 30)
( 0
30
0.537446
0.90758
0. 9
) 0 242
X
p X
p X
p X

 
 




7. 7
Normal Distribution Curve:
Note: I used my previous sample data (Normal distribution) to create this curve.
1- (22 27)
p X
  Red Area
2- ( 26)
p X  Red Area
3- ( 30)
p X  Green Area
( 30)
p X  Red Area

8. 8
Summary and Learning Outcomes:
Random variable: Is any function that assigns a numerical value to each possible outcome.
Classes of Random Variables
Discrete random variables: a discrete random variable can take one of a countable list of
distinct values.
 An example of a discrete random variable is the number of people with type O blood in
a sample of ten individuals. The possible values are 0, 1, 2, …., 10 a list of district
values.
Continuous random variables: a continuous random variable can take any value in an interval
or collection of intervals.
 An example of a continuous random variable is height for adult women. With accurate
measurement to any number of decimal places, any height is possible within the range of
possibilities for heights.
General equation for binomial random variable and finding probability for them:
Where:
P(x) = binomial distribution probability function.
x = number of successes desired.
𝑥0 = adjusted number of successes for the discrete random variable.
The formula for b (x; n, p) is made up of two parts:
1.The first part, 𝑛𝑥, gives the number of simple events in the sample space (consisting of all
possible listings of successes and failures in n trails).
2.The second part, (1 )n x
p 
 , gives the probability for each of the simple events for which x,
multiply p for each success and (1- p) for each failure to get (1 )n x
p 
 . This probability distribution
is called the binomial distribution because for x = 0, 1, 2, ……, and n. The values of the
probabilities are the successive terms of the binomial expansion of [𝑃+(1−𝑃)𝑛 ]: for the same
reason, the combination quantities
n
x
 
 
 
are referred to as binomial coefficients. Actually, the
preceding equation defines a family of probability distribution, with each members characterized
by a given value of the parameter (P) and the number of trails (n).

9. 9
The mean and the variance of a probability distribution:
Besides the binomial distribution, there are many other probability distributions that have
important engineering applications. However, before we go any further, let us discuss some general
characteristics of probability distributions. Another important such characteristics, that of the
symmetry or skewness of a probability distribution. Another distinction is that the histogram of the
second distribution is broader, and we say that the two distributions differ in variation. The mean
of a probability distribution is simply the mathematical expectation of a random variable having
that distribution.
Mean: .
Standard deviation: . ( ) . (1 )
n p
S D S n p p
 
  
Where:
n = number of trials
P = probability of success
1 – p = probability of failure
Poisson distribution
variance =
Standard deviation
.
( )
!
. ( )
x
S D S
P x
e
x

 


 



Lambda = np
Lambda = expected number of success.
Note: an acceptable rule of thumb is to use this approximation If n > 20 and P< 0.05 If n > 100, the
approximation is generally excellent as long as (np) <10.
Normal distribution: this is a mathematical expression which may be used to describe the probable
outcomes of certain processes. A statistical test for the accuracy of this assumption is treated later.
What is a confidence interval? A confidence interval measures the probability that a population
parameter will fall between two set of values. The confidence interval can take any number of
probabilities, with the most common being 95% or 99%.
Confidence interval: in survey sampling, different samples can be randomly selected from the
same population; and each sample can often produce a different confidence interval. Same
confidence intervals include the true population parameter; other does not.

10. 10
The following basic properties are used (see figure 6-2)
1. The normal distribution is symmetrical about the mean.
2. The total area under the normal distribution curve is equal to 1% or 100%.
3. The area under the curve between μ + σ and μ - σ is 0.6827.
4. The area under the curve between μ + 1.96σ and μ - 1.96σ is
0.9500.
5. The area under the curve between μ + 2σ and μ - 2σ is 0.9545.
6. The area under the curve between μ + 3σ and μ - 3σ is 0.9971.
7. The area under the curve between μ + and μ - is 1.000
Useful probability relationships for normal distribution:
1.P (z < a)
2.P (z > a)
3.P ( a < z < b)
4.P (z < μ - d) = p (z > μ+d)
Note: A normal random variable with mean μ = 0 and standard deviation = 1 is said to be a standard
normal random variable and to have a standard normal distribution.
Note: To find the area under the standard normal curve, use Table A.1 for (z ≤ 0) and (z ≥ 0) instead
of Equation. Because Equation is difficult to calculate the area.

11. 11
Discussion:
KEY TAKEAWAYS
 A random variable is a variable whose value is unknown or a function that assigns
values to each of an experiment's outcomes.
 Random variables appear in all sorts of econometric and financial analyses.
 A random variable can be either discrete or continuous in type.
My Results are O.K in Binomial and Poisson distribution and I calculated by modern
calculater and checked by Table but.
In normal distribution result is Significant and:
In the first probability area is equal to 0.537 Falls in the middle it is Red Zone.
In the Second probability area is equal to 0.5 Falls in the Left side it is Red Zone.
In the Third probability area if ( 30)
p X  is equal to 0.907 in the Right side it is Green Zone.

12. 12
Reference:
1. https://www.britannica.com/science/statistics/Random-variables-and-probability-
distributions
2. https://www.investopedia.com/terms/r/random-
variable.asp#:~:text=A%20random%20variable%20is%20a,each%20of%20an%20experime
nt's%20outcomes.&text=Random%20variables%20are%20often%20used,statistical%20rela
tionships%20among%20one%20another.
3. https://en.wikipedia.org/wiki/Random_variable
4. https://www.toppr.com/guides/maths/probability/random-variable-and-its-probability-
distribution/
5. Jessica M utts,2010, Mind on Statistics , University of California, Irvine, 4th
edition.
6. Prof.Dessouky, Engineering Statistics, ISE-180.
7. Engineering Statistics 2019-2020 Lecurer , Ms.Dilveen H. Omar.