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## Proof related to Axioms of Probability

These problems and Proofs are adapted from the textbook: Probability and Random Process by Scott Miller 2ed Problem-1: Proof that for events A and B the following holds: $$Pr\left(A\cup B\right) = Pr\left(A\right)+Pr\left(B\right)-Pr\left(A\cap B\right)$$ Solution: First, note that the sets $$A\cup B$$ and $$B$$ can be rewritten as $$A\cup B=\{A\cup B\}\cap\{\overline{B}\cup B\}=\{A\cap\overline{B}\}\cup B$$ $$B=B\cap\{A\cup\overline{A}\}=\{A\cap B\}\cup\{\overline{A}\cap B\}.$$ Hence, $$A\cup B$$ can be expressed as the union of three mutually exclusive sets. $$A\cup B=\{\overline{B}\cap A\}\cup\{A\cap B\}\cup\{\overline{A}\cap B\}$$ Next, rewrite $$\overline{B}\cap A$$ as $$\overline{B} \cap A= \{\overline{B}\cap A\}\cup\{\overline{A}\cap A\}= A\cap\{\overline{A}\cup\overline{B}\} = A\cap\{\overline{A\cap B}\}$$ Likewise, $$\overline{A}\cap B=B\cap\{\overline{A\cap B}\}$$. Therefore $$A\cup B$$ can be rewritten as the following...

## Introduction to Probability

In most of the engineering career, we encounter some signals in the form of noise, disturbance, interference, etc. As an engineer in fact as control or communication engineer, we need to tackle these signals so they won’t cause a problem to our original system. But how to identify such signals the answer lies in the theory of probability, we classify such signals as random processes and they normally follow some distribution like Gaussian/Normal, Exponential, Uniform etc. Before jumping into this theory I would like you to understand some basic concept of Probability. Some definitions of probability theory are given below:...

## Gram-Schmidt Orthogonalization Procedure

In Digital communication, we apply input as binary bits which are converted into symbols and waveforms by a digital modulator. These waveforms should be unique and different from each other so we can easily identify what symbol/bit is transmitted. To make them unique, we apply Gram-Schmidt Orthogonalization procedure. Now consider that we have a waveform $$s_1(t)$$ and we assume that its energy is $$\varepsilon_1$$. Then we can construct our first waveform as: $$\begin{equation} \psi_1(t) = \frac{s_1(t)}{\sqrt{\varepsilon_1}} \label{energy_1} \end{equation}$$ So now we have our first waveform which has energy = 1. Now we have our second waveform available known as...

## Comparison of 3G and 4G Cellular Services

Mobile communication is one of the hottest areas and it is developing extremely fast in present times, thanks to the advances of technology in all the fields of mobile and wireless communications. Nowadays the use of 3G mobile communication systems seem to be the standard, while 4G stands for the next generation of wireless and mobile communications. 3G is currently the world’s best connection method when it comes to mobile phones, and especially for mobile Internet. 3G stands for 3rd generation as it just that in terms of the evolutionary path of the mobile phone industry. 4G means 4th...