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Sub Chapter 2.5

Group V

II.5 Other Distribution And Density Examples

Binomial

Poisson

Uniform

Exponential

Rayleigh

Some example of other distribution function :

Discrete Random variables

Continuous Random Variables

Group V2.5.1 Binomial

Let 0 < P < 1 , and N = 1,2,… so the function

Called binomial density function and it applies to many games of chance, detection problems in radar & sonar, and many experiment having only two posible outcomes on any giveb trial.

is binomial coefficient

By integration we find :

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Ilustration binomial density & distribution function for N=6 and p=0.25

Binomial Density Distribution

Binomial Density function for case

Group V2.5.2 Poisson

Poisson Random Variable X has a density and distribution given by

Where b > 0. Applies to a wide variety of counting type application. It describes the number of defective units in a sample taken from a production line, the numbers of telephone calls made during a period of time, the number of electron emitted from a small section of a cathode in a given time interval.

By integration we find :

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Example :Assume automobile arrivals at a gasoline station are poisson and occur at an average rate of 50/h. The station has only one gasoline pump. If all cars are assumed to require one minute to obtain fuel, what is the probability that a waiting line occur at the pump ?

Answer :A waiting line will occur if two or mor cars arrive in any one-minute interval. The probability of this event is one minus the probability that either none or car arrives. From (III) equation with =(50/60) car minutes & T=1 minute, we have b = 5/6, and by using equation (II) ……

If the time interval of interest has duration T, and the events being counted are known to occur at an average rate and have a poisson distribution, then …….

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Expect a line at the pump about 20.32% of the time

Group V2.5.3 Uniform

Uniform probability density and distribution function are defined by :

For real constant - < a < . Applied for the quantization of signal samples prior to encoding in digital communication systems.

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Uniform probability density function

Distribution function

Group V2.5.4 Exponential

The exponential density and distribution function are defined by :For real numbers - < a < and b > 0Useful in describing raindrop sizes when a large number of rainstorm measurement are made & also describe fluctuations in signal strength received by radar from aircraft.

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Example :The power reflected from an aircraft of complicated shape that is received by a radar can be described by an exponential random variable P. The density of P is therefore :Where Po is the average amount of received power. At some given time P may have a value different from its average value and we ask : What is the probability that received power is larger than the power received on the average ?

Answer : - We Must find P{P>P0} = 1 – P{P P0} = 1 – Fp(P0)

In other word, the received power is larger than its average value about 36,8 % of the time

Group V2.5.5 Exponential

The exponential density and distribution graph :

Exponential probability density distribution

Exponential probability density functions

Group V2.5.5 Rayleigh

The Rayleigh density and distribution function are defined by :

For real constants - < a < and b > 0Describes the envelope of one type of noise when passed through a bandpass filter. It is also is important in analysis of errors in various measurement systems.

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Example :We find the value x = x0 of a rayleigh random variable for which P{ X X0 } = P { X0 < X }. This value of X is called the median of random variable. The probability condition requires P{ X X0 }= Fx(x0) = 0.5.From equation we find Fx(x0) = 1 – exp [- (xo – a)2 /b] = 0.5The solution for X0 follows the natural logarithm. We find X0 = a + [b ln(2)]2.The Median is similarly defined for random variables other than Rayleigh ; it is the value of X for which the probability is 0.5 that values of X do not exceed the median.

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