Let X Denote the Number of Brand X VCRs Sold$$\begin{array}{l}
\begin{array} { l l l l l l }
\hline x & 0 & 1 & 2 & 3 & 4 \
\hline & .1 & .2 & .3 & .25 & .15
\end{array}\
p _ { x } ( x )\
\hline
\end{array}$$

Question

Let X Denote the Number of Brand X VCRs Sold$$\begin{array}{l}
\begin{array} { l l l l l l } 
\hline x & 0 & 1 & 2 & 3 & 4 \
\hline & .1 & .2 & .3 & .25 & .15
\end{array}\
p _ { x } ( x )\
\hline
\end{array}$$

Examlex · Accepted Answer

The Answer is a.

p ( 4,2 ) = P ( Y = 2 \mid X = 4 ) \cdot P ( X = 4 ) = \left[ \left( \begin{array} { l } 4 \2\end{array} ight) ( .7 ) ^ { 2 } ( .4 ) ^ { 2 } ight] \cdot ( .15 ) = .0397

b.

P ( X = Y ) = p ( 0,0 ) + p ( 1,1 ) + p ( 2,2 ) + p ( 3,3 ) + p ( 4,4 ) = .1 + ( .2 ) ( .7 ) + ( .3 ) ( .7 ) ^ { 2 } +

( .25 ) ( .7 ) ^ { 3 } + ( .15 ) ( .7 ) ^ { 4 } = .5088

c.

p ( x , y ) = 0 ext { unless } y = 0,1 , \ldots , x ; x = 0,1,2,3,4

For any such pair,

p ( x , y ) = P ( Y = y \mid X = x ) \cdot P ( X = x ) = \left( \begin{array} { l } x \y\end{array} ight) ( 7 ) ( .3 ) ^ { x -y} \cdot p _ { x } ( x )

p_{y} ( 4 ) = p ( y = 4 ) = p ( x = 4 , y = 4 ) = p ( 4,4 ) = ( .7 ) ^ { 4 } \cdot ( .15 ) = .0360

p_{y} ( 3 ) = p ( 3,3 ) + p ( 4,3 ) = ( .7 ) ^ { 3 } ( .25 ) + \left( \begin{array} { l } 4 \3\end{array} ight) ( .7 ) ^ { 3 } ( .3 ) ( .15 ) = .1475

p _{y} ( 2 ) = p ( 2,2 ) + p ( 3,2 ) + p ( 4,2 ) = ( .7 ) ^ { 2 } ( .3 ) + \left( \begin{array} { l } 3 \2\end{array} ight) ( .7 ) ^ { 2 } ( .3 ) ( .25 )

+ \left( \begin{array} { l } 4 \2\end{array} ight) ( .7 ) ^ { 2 } ( .3 ) ^ { 2 } ( .15 ) = .2969

P_{y} ( 1 ) = p ( 1,1 ) + p ( 2,1 ) + p ( 3,1 ) + p ( 4,1 ) = ( .7 ) ( .2 ) + \left( \begin{array} { l } 2 \1\end{array} ight) ( .7 ) ( .3 ) ( .3 )

\left( \begin{array} { l } 3 \1\end{array} ight) ( .7 ) ( .3 ) ^ { 2 } ( .25 ) + \left( \begin{array} { l } 4 \1\end{array} ight) ( .7 ) ( .3 ) ^ { 3 } ( .15 ) = .3246

p , ( 0 ) = 1 - [ .3246 + .2969 + .1475 + .0360 ] = .1950

Let X Denote the Number of Brand X VCRs Sold

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