Download p(x)

(ii) Poisson distribution
•The Poisson distribution resembles the binomial distribution if
the probability of an accident is very small.
•Some events are rather rare, they don't happen that often. Still,
over a period of time, we can say something about the nature of
rare events.
•The Poisson distribution is most commonly used to model the
number of random occurrences of some phenomenon in a
specified unit of space or time.
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(ii) Poisson distribution
Let p become very small, n very large, but
finite. then
r   rp( r )  np
e  r ( r )r
p( r ) 
r!
which depends on r only
Mean value
r   rp( r )  np
Variance  2  np
The Poisson distribution applies to many physical phenomena,
such as radioactive decay, in which n (the number of atoms) is
very large but p (the probability of a given atom decaying in a
given time) is very low. …CY1B2 Statistics
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Example: Over a given period there are 10 6 cars on the road,
and the probability of a given car having an accident is 10 -4.
Find the mean number of accidents , and variance.
Mean value
Variance
r   rp( r )  np  10 10  100
6
4
  np  r  100.
2
The standard deviation is then 10. This means a 10% rate in the
accident rate over a given period is not significant.
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Discrete Distribution: A statistical distribution whose
variables can take on only discrete values.
The mathematical definition of a discrete probability
function, p(x), must satisfies the following properties.
1.The probability that x can take a specific value is p(x).
That is
p [ X  x j ]  p( x j )  p j
2. p(x) is non-negative for all real x.
3. The sum of p(x) over all possible values of x is 1.

p( x j )  1
j
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Continuous Distribution: A statistical distribution for which
the variables may take on a continuous range of values. The
mathematical definition of a continuous probability
function, f(x), satisfies the following properties:
1. The probability that x is between two points a and b is
b
p[ a  x  b ]   f ( x )d ( x )
a
2. It is non-negative for all real x.
3.
The integral of the probability function is one, that is



f ( x )d ( x )  1
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What does this actually mean?
Since continuous probability functions are defined for an
infinite number of points over a continuous interval, the area
under the curve between two distinct points defines the
probability for that interval.
(Probabilities are measured over intervals, not single points.
The probability at a single point is always zero.)
The height of the probability function can in fact be greater than
one. The property that the integral must equal one is equivalent
to the property for discrete distributions that the sum of all the
probabilities must equal one.
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• Continuous probability distributions
(i) Probability density
Suppose we sample a waveform as shown in above figure,
with a digital voltmeter, taking a reading with a resolution δx
(e.g. =0.1V), over a period of time t.
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In order to intuitively develop the concept of continuous probability
function, we uses the method of histogram, and then see how to
define continuous probability function from histogram.
A histogram is obtained by splitting the range of the data into equalsized bins (called classes). Then for each bin, the number of points
from the data set that falls into each bin is counted. That is
•Vertical axis:
Frequency (i.e., counts for each bin)
•Horizontal axis: Response variable
The histogram has an additional variant whereby the counts are
replaced by the normalized counts. The names for these variants are
the relative histogram and the relative cumulative histogram.
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A histogram as shown above gives the number of reading n
versus range x.
nx
n
The right hand side plots
versus x, where n is the total
number of reading, when n becomes very large.
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The first step normalized
count is the count in a class
divided by the total number of
observations. In this case the
relative counts are normalized
to sum to one (or 100 if a
percentage scale is used). This
is the intuitive case where the
height of the histogram bar
represents the proportion of
the data in each class. Or the
probability of the data falling
into each range (each class).
Sums up to one.
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To get a smoother curve, reduce δx - 0.1v,0.01v,0.001v,
0.0001v- 0, such that each range becomes infinitely small,
but the problem is that data points in every class nx/n 0,
becomes negligible or  0.
2. The second step normalized count is the count in the class
divided by the number of observations times the class width.
For this normalization, the area (or integral) under the
histogram is equal to one.
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The resultant smooth curve is a plot of probability
density function p(x)
nx 1
p( x )  lim
n   n x
x  0
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Clearly p(x)δx is the probability of a reading being in a small
range δx .
The probability of a reading between two values
x2
p( x1  x  x2 )   p( x )dx
x1
The area under the curve is unity.



p( x )dx  1
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The expressions for the mean, mean square and variance for
continuous probability distribution p(x) are

x   xp( x )dx


x   x p( x )dx
2
2


   ( x  x ) 2 p( x )dx  x 2  ( x ) 2
2

The mode: value of x that corresponds to maximum p(x)
The Median: value of x that divides plot into equal areas.
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Example: Deduce the probability density for the sawtooth
waveform.
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It can be deducted that the wave form moves
all range of [0,A].

p( x )  




uniformly through
0 x A
1
A
x  0, or,
0
p( x )dx  
A
0
x A
1
1
dx   A  1
A
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