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chapter3_exercise.lyx
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chapter3_exercise.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\begin_document
\begin_header
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\begin_body
\begin_layout Subsubsection*
3.9 Functions of two or more Random variables
\end_layout
\begin_layout Standard
9.
\begin_inset Formula $P(Y_{1}\le0.1,Y_{n}\leq0.8)=P(Y_{1}\leq0.1|Y_{n}\leq0.8)P(Y_{n}\leq0.8)$
\end_inset
\end_layout
\begin_layout Standard
First,
\begin_inset Formula $P(Y_{n}\leq0.8)=0.8^{n}$
\end_inset
\end_layout
\begin_layout Standard
Second,
\begin_inset Formula $P(Y_{1}\leq0.1|Y_{n}\leq0.8)$
\end_inset
is conditional probability, we need to compute event
\begin_inset Formula $Y_{1}\leq0.1$
\end_inset
if know
\begin_inset Formula $Y_{n}\leq0.8$
\end_inset
.
More details, we already known that
\begin_inset Formula $Y$
\end_inset
is uniform distribution in
\begin_inset Formula $[0,1]$
\end_inset
.
But it we know
\begin_inset Formula $Y_{n}orY_{max}\leq0.8$
\end_inset
, then we can sure that
\begin_inset Formula $(Y|Y_{max}\leq0.8)$
\end_inset
now is uniform distribution in
\begin_inset Formula $(0,0.8),$
\end_inset
so
\begin_inset Formula $P(Y\leq0.1|Y_{max}\leq0.8)=1/8$
\end_inset
.
So
\begin_inset Formula $P(Y_{1}\leq0.1|Y_{n}\leq0.8)=(1-(\frac{7}{8})^{n}).0.8^{n}=0.8^{n}-0.7^{n}$
\end_inset
.
\end_layout
\begin_layout Paragraph*
\series medium
15.
We have:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P(X_{1},...,X_{n})=P(X_{1})...P(X_{n});Y_{1}=r_{1}(X_{1}),...,Y_{n}=r_{n}(X_{n})
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $P(Y_{1}...Y_{n})=\sum_{x_{1},..,x_{n}:Y_{1}=r_{1}(x_{1}),...,Y_{n}=r_{n}(x_{n})}P(x_{1}...x_{n})=\sum_{x_{1},..,x_{n}:Y_{1}=r_{1}(x_{1}),...,Y_{n}=r_{n}(x_{n})}P(x_{1})...P(x_{n})=\prod_{i}\sum_{x_{i}:Y_{i}=r_{i}(x_{i})}P(x_{i})=\prod_{i}P(Y_{i})$
\end_inset
.
\end_layout
\begin_layout Standard
And with
\begin_inset Formula $Y_{1},...,Y_{j}(0<j\le n)$
\end_inset
, we similarly prove that
\begin_inset Formula $Y_{1},...,Y_{j}$
\end_inset
is independence.
\end_layout
\begin_layout Standard
So
\begin_inset Formula $Y_{1},...,Y_{n}$
\end_inset
is independence.
\end_layout
\end_body
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