Q2
← Back
Basic Info
Probability
└── Homework
└── W5
└── Q2.tex
Preview
\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage[margin=1in]{geometry}
\usepackage{fancyhdr}
\usepackage{enumerate}
\usepackage[shortlabels]{enumitem}
\pagestyle{fancy}
\fancyhead[l]{Li Yifeng}
\fancyhead[c]{Homework \#5}
\fancyhead[r]{\today}
\fancyfoot[c]{\thepage}
\renewcommand{\headrulewidth}{0.2pt}
\setlength{\headheight}{15pt}
\newcommand{\bP}{\mathbb{P}}
\begin{document}
\section*{Question 2}
\noindent Consider a binary communication channel, with every digit in the input having a Bernoulli distribution with parameter $p = 0.8$ (i.e., the probability of sending $1$ is $p$). A "word" contains 6 digits: $X_1$, $X_2$, ..., $X_6$.
\bigskip
\begin{enumerate}[start=1,label={\bfseries Part \arabic*:},leftmargin=0in]
\bigskip\item What is the probability that a word contains exactly four $1$'s and two $0$'s?
\subsection*{Solution}
As given, we can define the Bernoulli distribution with sample space $\{0,1\}$ ($0$ denotes $not\enspace received$ and $1$ denotes $received$) and with parameter $p = 0.8$, then we can define its Binomial distribution as
\[X\sim \text{Bin}(n,p),\quad n = 6\]
Then we can know the probability that a word contains exactly four $1$'s and two $0$'s
\[
\begin{aligned}
\bP(X = 4) &= \dbinom{n}{4}p^4(1-p)^{n-4}\\
&= 0.245\,76
\end{aligned}
\]
\subsection*{Answer}
\[\boxed{\bP(X = 4) = 0.245\,76}\]
\bigskip\item What is the probability that a word contains at least four $1$'s?
\subsection*{Solution}
\[
\begin{aligned}
\bP(X\ge 4) &= \sum_{k = 4}^n\bP(X = k)\\
&= \sum_{i = 4}^n\dbinom{n}{k}p^k(1-p)^{n-k}\\
&= 0.901\,12
\end{aligned}
\]
\subsection*{Answer}
\[\boxed{\bP(X\ge 4) = 0.901\,12}\]
\bigskip\item Assume that the first digit is $X_1 = 1$. What is the probability that the sum of the first two digits is $2$?
\subsection*{Solution}
Let the trials be denoted as
\[X_i,\quad i\in\{1,2,\dots,n\}\]
Then we have
\[
\begin{aligned}
\bP_{X_1=1}(X_1 + X_2 = 2) &= \bP(X_2 = 1)\\
&= p\\
&= 0.8
\end{aligned}
\]
\subsection*{Answer}
\[\boxed{\bP_{X_1=1}(X_1 + X_2 = 2) = 0.8}\]
\end{enumerate}
\end{document}