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Chapter VI – Statistical Analysis of Experimental Data
6.3.2. Some typical probability distribution functions
6.3.2.a. Binomial
distribution
This distribution is restricted to random variables that can only have two possible
outcomes: success (or acceptable) or fail (or unacceptable). The following conditions
have to be satisfied in order to apply a binomial distribution:
1- Only two possible outcomes.
2- The probability of success remains constant throughout the experiment.
3- The experiment consists of n independent trials.
The number (r) of success in a total of (n) trials is then given by:
n
nr
P(r )    p r 1  p 
r 
n
n!
With   
is n combination r.
 r  r!n  r  !
We can get the expected number of successes in n trials for a binomial distribution as:
  np
And standard deviation for the binomial distribution of:   np1  p 
Now, if your objective is to get the probability of an event happening less or equal a
certain number of times ( k  n ), this is given as:
k
k
n
ni
P(r  k )   P(r  i )     p i 1  p 
i
i 0
i 0  
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Chapter VI – Statistical Analysis of Experimental Data
6.3.2.b. Poisson
distribution
It is used to estimate the number of random occurrences of an event in a specified
interval of time or space if the average number of occurrences is already known.
Her are some conditions satisfied by Poisson distribution:
1- The probability of occurrence of any event is the same for two intervals of the
same length.
2- The probability of the occurrence of an event is independent of the occurrence
of other events.
P( x) 
e   x
x!
 is the expected (or mean) number of events during the interval of interest.
E (x)    
and   
To determine the probability of a certain number of events is lower than k:
k
P( x  k )  
i 0
Instrumentation and Measurements \ LK\ 2009
e   i
i!
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Chapter VI – Statistical Analysis of Experimental Data
6.3.2.c. Normal
distribution (Gaussian distribution)
It is used when the values obtained are due uniquely to random factors and occurrences
of both positive and negative deviations are equally probable.
f ( x) 
2
2
1
e  x  / 2
 2
Where: x is the random variable;  is the population mean and  is the population
standard deviation.
Figure 6.3. Gaussian (normal) distribution.
For a Gaussian distribution, the probability of having a single value x between two
values x1 and x2 is:
x2
x2
x1
x1
P( x1  x  x2 )   f ( x) dx  
2
2
1
e  x  / 2 dx
 2
Since this integral can not be solved analytically, it is performed numerically. For this
purpose, a change of variable will make our life easier.
z
Leading to: f ( z ) 
x

1 z2 / 2
for  =1. This function is called the standard normal
e
2
density function.
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Chapter VI – Statistical Analysis of Experimental Data
Figure 6.4. Standard normal density function.
We can write now the above integral in terms of z:
z2
P( x1  x  x2 )  P( z1  z  z 2 )   f z  dz
z1
The probability that a measurement will fall in one of more standard deviations is
summarized below:
Confidence Interval
1
2
3
 3.5 
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Confidence Level
68%
95.44%
99.74%
99.96%
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Chapter VI – Statistical Analysis of Experimental Data
Example
Automobiles visit inspection station an average of 40 times in an eight-hour workday.
What is the probability that more than five automobiles will visit the inspection station in
a one-hour period?
Example
The lighting department of a city has installed 2000 electric lamps with an average life of
1000 h and a standard deviation of 200 h.
a) after what period of lightning hours would you expect 10% of the lamp to fail?
b) how many lamps may be expected to fail between 900 and 1300 h?
Example
An electronic production line yields 95% satisfactory parts. Assume that the quality of
each part is independent of the others. Four parts are tested. Determine the probability
that:
a) all parts tested are satisfactory.
b) At least two parts are satisfactory.
6.4. Parameter estimation
The main principle behind statistical methods is use a small sample and extract from it
information on the population. However, the sample is usually small and as a
consequence different samples might lead to different means and standard deviations
for the population. It is, therefore, essential to determine the uncertainty of the
parameters extracted from the sample.
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