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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:...