What are Random Variables Probability And Stochastic Process?

Hello everyone, I hope that you all are doing good in your lives. The topic for today is probability and stochastic process. In this blog, we resume studying the development of background materials necessary for a better understanding of communication systems. So we will be referring to the statistical character of a random signal; which acts as a pillar to the communication theory. Thus for studying random variables, we need to learn about their mathematical discipline i.e. probability. and will also be going through a stochastic process

Introduction to Probability and stochastic process

Examples of a random signal are seen in every practical system. We say a signal to be random when it is not possible to predict its accurate value in advance. For example, consider a radio communication system. The signal at the receivers in such type of systems consist of three parts; Firstly the information component, secondly the receiver noise, and lastly the random signal. The information-bearing signal component may be a voice signal that typically consists of randomly spaced bursts of energy. The interference component may represent fakely produce electromagnetic waves produced by the other communication systems operating in the vicinity of the radio receiver. A source of receiver noise is thermal noise, which is due to the random motion of electrons in the conductor device at the front end.

The mathematical discipline that deals with the statistical character of a random signal are probability.

Probability and stochastic process

Probability

The theory of probability is in which explicitly or implicitly the outcome of the experiment can be obtained as a subject of chance. Moreover, if the experiment is one after the other or again after an interval the output may differ from the last one. This is because of the underlying random character of the chance mechanism. Such experiments go by the name random experiment. For example, the experiment may be the observation of the result of tossing a fair coin. In this experiment, the possible trail of outcome is head or tail.

There are mainly two approaches to the definition of probability;

1. Realative frequency of occurrance
2. set theory

Relative frequency of occurance

According to this theory in n trials of a random experiment, if we expect an event A to occur m times, then we assign the probability m/n to the event A. This definition of probability is straightforward and simple to apply However, the stochastic process is somewhat difficult to understand.

However, there are situations where experiments are not repeated in this case we use the second type i.e. set theory

Set theory

It is one of the general definitions where the output is dependent on the set theory and a set of related mathematic axioms. Whenever we perform a random experiment; it is natural for us to be aware of the various outcome that is likely to arise. If an experiment has K possible outcomes, then for Kth possible outcome we have a point called sample point, which we denote by S_k.

The sample space S may be discrete with a countable number of outcomes, for example, rolling a die. Besides the sample space (set of all possible outcomes of an experiment) may be continuous such as the voltage measurement.

A probability measure P is a function that assigns a non-negative number to an event A in the sample space S and satisfies the points below;

1. 0 ≤ P[A] ≤ 1
2. P[S] = 1
3. The two events A and B are two exculsive event, then P[A ∪ B] = P[A] + P[B]

Conditional Probability

Suppose we perform an experiment that involves a pair of events A and B. Let P[B][A] represents the probability of event B given that event A has occurred before. The probability P[A][B] is the conditional probability of B given A. Assuming that A has a nonzero probability, the conditional probability P[B][A] is define as;

   

When P[A∩B] is the joint probability of A and B. We may write the above equation as;

   

We can also write it as;

   

Accordingly, we can state that the joint probability of two events expresses the product of the conditional probability of one event given the other, and the elementary probability of others.

Situations may exist where the conditional probability P[A][B] and the probabilities P[A] andP[B] are easily determinable directly, but the probability P[B][A] is desirable. With the condition that P[A] ≠ 0, we may determine P[B][A] by using the relation

   

since P[B|A] = P[B]

   

   

   

That is the conditional probability of event A, assuming the occurrence of event B, is simply equal to the elementary probability of event A. We thus see that in this case, a knowledge of the occurrence of one event tells us more about; the probability of occurrence of the other event than we knew without that knowledge. event A and B that satisfy this condition are said to be statistically independent.

Theme example – Stochastic Model

One situation where probability theory and random processes play an important role in communications is in the analysis of mobile radio performance. We use the term mobile radio to encompass indoor and outdoor forms of wireless communication; where a radio transmitter or receiver is capable of being moveable, regardless it actually moves or not. Due to the complex and variable nature of the mobile radio channel, it is not feasible to use a deterministic approach for its characterization. Rather, it is necessary to resort to the use of measurements and statistical analysis.

The major propagation problems encounter in the use of cellular radio in built-up areas are due to the fact that the antenna of a mobile unit may lie well below the surrounding buildings. In simple words, there is no line-of-sight path to the ground station. Instead, radio propagation takes place mainly by way of scattering from the surface of the surrounding buildings and by diffraction around them. The important point to note is that the energy reaches that receiving antenna via more than one path. Accordingly, we speed up a multipath phenomenon in that various incoming radio waves reach their destination from a different direction and with different time delays.

Nature

However, to understand the nature of the multipath phenomenon; consider first a static multipath environment involving a stationary receiver and a transmit signal that consists of a narrow-band signal. Let assume that two attenuated version of the transmitted signal. The effect of the differential time delay is to introduce a relative phase shift between the two components of the signal received. We may then identify one of two extreme cases that can arise:

1. The realative phase shift is zero, in which case the two component add constructively.
2. The relative phase shift is 180 degree, in which case the two components add destructively.

Between these two extreme cases are a variety of situations where we may obtain partial constructive or destructive interference.

Note:- That the relative phase shift of the two signals will vary with the position since the relative time delay also varies with position.

The net result is that the envelope of the receiver signal varies with position in a complicated fashion, shown by the experimental record of the received signal. The figure clearly displays the fading nature of the receiving signal it is measure in dBm. The unit dBm is define as 10log10(P/Po), with P denoting the power being measured and Po = 1 milliwatt. P is the instantaneous power of the receiving signal.

Conclusion

However, here we are at the end of the blog. I hope that you are satisfied with the content above. And have got answers to all of your questions. If you do like the blog then please mention the part that you like the most. Besides if you do have any doubts regarding the topic then please feel free to mention below or contact us. We would grateful to have any suggestions on the topic that you like to read next on.

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