Random Variables and Stochastic Processes – 0903720

Dr. Ghazi Al SukkarEmail: ghazi.alsukkar@ju.edu.joOffice Hours: will be posted soon

Course Website:http://www2.ju.edu.jo/sites/academic/ghazi.alsukkar

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Most common used RV.s• Continuous-Type:

– Gaussian– Log-Normal– Exponential– Gamma– Erlang– Chi-square– Reyleigh– Nakagami-m– Uniform

• Discrete-Type:– Bernoulli– Binomial– Poisson – Geometric– Negative Binomial– Discrete Uniform

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Gaussian (or Normal) Random Variable :

• This is a bell shaped curve, symmetric around the parameter and its distribution function is given by

(Tabulated)

Since depends on two parameters and the notation is used to denote a Gaussian RV.

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• : Standard Normal RV: zero mean and Unity variance.

• Most important and frequently encountered random variable in communications.

𝜇 𝜇

Large Small

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Log-normal Distribution

• If is a random variable with a normal distribution, then has a log-normal distribution.

• Likewise if is log-normal distribution, then is normal distribution. Denoted as

• .

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Exponential distribution

• The exponential distribution represents the probability distribution of the time intervals between successive Poisson arrivals.

• is exponential if:

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The Memoryless property of Exponential Distribution

• The exponential distribution is without memory.• The exponential distributions is the unique

continuous memoryless distributions.• Let

• Let represents the lifetime of an equipment, if the equipment has been working for time , then the probability it will survive an additional time depends only on , and is identical to the probability of survival for time of a new equipment.

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Example

• The amount of waiting time a customer spends at a restaurant has an exponential distribution with a mean value of 5 minutes.

• The probability that a customer will spend more than 10 minutes in the restaurant is:

• The probability that the customer will spend an

additional 10 minutes in the restaurant given that he has been there for more than 10 minutes is:

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Gamma (Erlang) Distribution

• Denoted by , .• is the Gamma function, for integer.

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• ,

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Erlang Distribution

• Erlang Distribution is a special case of Gamma distribution where the shape parameter is an integer. It is

• Let ,

Put .Application: The number of telephone calls which might be made at the same time to a switching center.

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CHI-Square Distribution

• , , • It is a special case of Gamma distribution when , and

Chi-square with degree of freedom.

• If : will obtain an exponential distribution.

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Rayleigh Distribution• is Rayleigh distribution with parameter .• , . • , • Application: used to model attenuation of wireless signals facing

multi-path fading.

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Nakagami-m Distribution

• A generalization of Rayleigh distribution through a parameter .

• Put Rayleigh distribution• Application: gives greater flexibility to model

randomly fluctuating channels in wireless communication theory.

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Uniform Random Variable• A continuous random variable that takes values

between and with equal probabilities over intervals of equal length .

• The phase of a received sinusoidal carrier is usually modeled as a uniform random variable between

0 and . • Quantization error is also typicallymodeled as uniform.

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Discrete random variables

– Bernoulli– Binomial– Poisson – Geometric– Negative Binomial– Discrete Uniform

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Bernoulli Random Variable

• Simplest possible random experiment• Two possibilities:

– Accept/failure– Male/female– Rain/not rain

• One of the possibilities mapped to 1, , .

• Good model for a binary data source whose output is 1 or 0.• Can also be used to model the channel errors.

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Binomial Random Variable• is a discrete random variable that gives the number of 1’s in a sequence

of n independent Bernoulli trials.• , are statistically independent and Identically distributed (iid) Bernoulli

RV.s

0 2 4 6

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𝐹 𝑋 (𝑥 )

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Poisson Probability Mass Function

• Assume:1. The number of events occurring in a small time interval is as .2. The number of events occurring in non overlapping time intervals are independent.

• Then the number of events in a time interval have a Poisson Probability Mass Function of the form:

Where .Application:

– The number of phone calls at a call center per minute.– The number of time a web server is accessed per minute.

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Geometric distribution

• How many items produced to get one passing the quality control

• Number of days to get rain• Sequence of failures until the first success -

sequence of Bernoulli trials• Possible values:– If we count all trials – If we only count the failures

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Derivation of probability density function - counting all trials

• Let be the number of trials needed to the first success in repeated Bernoulli trials.

• Let's look at the sequence FFFFS with probability • A general sequence will be like FFF…FFS• The probability of having failures before the first

success is, • The cumulative distribution can be found to be

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The memoryless property

• What will happen to the distribution knowing that failures already occurred?

• That is we have been waiting for an empty cab and have experienced 7 occupied

• Formally

• That is, the probability of exceeding having reached is the same as the property of exceeding starting from the beginning. In other words no aging.

• Given that the first trials had no success, the conditional probability that the first success will appear after an additional trials depends only on and not on (not on the past).

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Negative Binomial Distribution

• Let be the number of Bernoulli trials required to realize success.

• If or fewer trials are needed for successes, then the number of successes in trials must be at least :

: Negative Binomial RV.: Binomial RV.

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• Let : the number of failures preceding the success.

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Uniform Probability Mass Function

1𝑛

𝑥1 𝑥2 𝑥3 𝑥𝑛

𝑓 𝑋 (𝑥)

𝑥