diff --git a/src/chapters/5/sections/normal/index.tex b/src/chapters/5/sections/normal/index.tex index f355c40e..27c4fa3e 100644 --- a/src/chapters/5/sections/normal/index.tex +++ b/src/chapters/5/sections/normal/index.tex @@ -1,5 +1,7 @@ \section{Normal} +\subsection{problem 24} +\input{problems/24} \subsection{problem 26} \input{problems/26} \subsection{problem 35} diff --git a/src/chapters/5/sections/normal/problems/24.tex b/src/chapters/5/sections/normal/problems/24.tex new file mode 100644 index 00000000..319eb45d --- /dev/null +++ b/src/chapters/5/sections/normal/problems/24.tex @@ -0,0 +1,19 @@ +Let $D$ be the event in which the woman gives birth on the due date. + +From the statement, $T \sim \mathcal{N}(0,8^2)$ days. +$T$ can be transformed to a standard Normal r.v. by the relation +$Z = (T-0)/8 = T/8 \sim \mathcal{N}(0,1)$. + +Assuming that the time $t$ on the timeline is in the unit of days, +the due date corresponds to the interval $t \in [0,1)$. +Therefore + +$$ +P(D) = P(0 \le T < 1) = P(0 \le 8Z < 1) = \Phi \left( \frac{1}{8} \right) - \Phi(0) +$$ + +$$ +P(D) = \Phi \left( \frac{1}{8} \right) - \frac{1}{2} +$$ + +\noindent where $\Phi(\cdot)$ is the CDF of the standard Normal distribution.