Steps for And give me full answer

Introduction and Background Material

$0\mathrm{.}1.$ Message transmission through a noisy communication channel

A digital message $"$M$"$ is created and sent through a noisy communication channel

$($Figure $1).$ The signal $"$ S $"$ to be transmitted consists of a series of zeros and ones:

symbol $"0"$ appears in the signal with probability $p_0$

symbol $"1"$ appears in the signal with probability $p_1=1-p_0$

The transmitted signal $"$ S $"$ is received at the other end of the channel as signal $"R".$

Due to noise in the communication channel, a transmitted bit may change during

transmission:

a transmitted bit $0$ may be received as $1$ with probability ${}_0($probability of

transmission error for symbol $0)$;

a transmitted bit $1$ may be received as $0$ with probability ${}_1($probability of

transmission error for symbol $1).$

The errors for different symbol transmissions are independent.

In order to create one bit of the transmitted message $"$S$"$ you have to:

Generate a random number: $m=$rand$()$

Generate the transmitted message $"S"$ as: $S=\{\begin{array}{ccc}0& if& m{p}_0\\ 1& if& m>p_0\end{array}$

In order to create the received signal $"R"$ you have to:

Generate a random number: $t=$rand$().$ Note that the random number $t$

should be different than the previous random number $m.$

Generate the received signal $"$R$"$ as: $R=\{\begin{array}{ccc}1& ifS=0\text{a}\text{n}\text{d}& t{}_0\\ 0& ifS=0\text{a}\text{n}\text{d}& t>{}_0\\ 1& ifS=1\text{a}\text{n}\text{d}& t>{}_1\\ 0& ifS=1\text{a}\text{n}\text{d}& t{}_1\end{array}$

Figure $1.$ Probabilities for symbol transmission error

Reference: "Introduction to probability", D$.$ Bertsekas and N$.$ Tsitsiklis, $2$nd

Edition, Athena Scientific, $2008.$

Probability of erroneous transmission

Consider the following experiment, where the required probabilities are given as:

$P_0=0\mathrm{.}40$;$0=0\mathrm{.}02,{}_1=0\mathrm{.}015$

You transmit a one$-$bit message $S$ and look at the received signal $R.$ If $RS,$

You repeat this experiment $N=10,000$ times and count the number of failures.

Find the probability that the transmitted bit will be received incorrectly, i$.$e$.$

the probability of failure.

SUBMIT the your answer and the Python code in a Word file. Use the table

below for your answer. Note: You will need to replicate the table in your Word

file, in order to provide the answer in your report. Points wil be taken off if you

do not use the table. bit $R.$ If $R=1,$ the experiment is a success, i$.$e$.$ success is defin

Conditional probability: $P(R=1|S=1)$

Consider the following experiment, where the required probabilities are given as

$0=0\mathrm{.}02,1=0\mathrm{.}015$

You create and transmit a one$-$bit message $S$ as you did before. The goal is to

calculate the conditional probability $P(R=1|S=1).$ This means that you will

focus only in those transmissions where $S=1.$

For all the events for which the transmitted signal is $S=1,$ look at the received

conditional event: $)=1|S|=(1$

You repeat this experiment $N=100,000$ times and count the number of successes.

Find the conditional probability $P(R=1|S=1),$ i$.$e$.$ the probability that if

you transmit the symbol $S=1,$ it will be received correctly.

SUBMIT your answer and the Python code in a Word file. Use the table below

for your answer. Note: You will need to replicate the table in your Word file, in

order to provide the answer in your report. Points will be taken off if you do not

use the table.

Conditional probability: $P(S=1|R=1)$

Consider the following experiment, where the required probabilities are given as:

$P0=0\mathrm{.}40$;$0=0\mathrm{.}02,1=0\mathrm{.}015$

You create and transmit a one$-$bit message S as you did before. The goal is to

calculate the conditional probability $P(S=1|R=1).$ This means that you will

only be interested in those messages where the received signal is $R=1.$

For all the events for which the received signal is $R=1,$ look at transmitted bit

$S.$ If $S=1,$ the experiment is a succes