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Magazine Chapter 1: Q6E (page 1)
Question:Suppose that a point (X,Y)is chosen at random from the regionSin thexy-plane containing all points (x,y) such thatx鈮�0,y鈮�0, and 4y+x鈮�4.
a. Determine the joint p.d.f. ofXandY.
b. Suppose that\({{\bf{S}}_{\bf{0}}}\)is a subset of the regionShaving area伪and determine\({\bf{Pr}}\left( {\left( {{\bf{X,Y}}} \right) \in {{\bf{S}}_{\bf{0}}}} \right)\).
Short Answer
- The Xand Y joint p.d.fis\({f_{\left( {X,Y} \right)}}\left( {x,y} \right) = \left\{ \begin{array}{l}\frac{1}{2}\;if\;\left( {x,y} \right) \in S\\0\;otherwise\end{array} \right.\)
- \(\Pr \left( {\left( {X,Y} \right) \in {S_0}} \right) = \frac{\alpha }{2}\)
Step by step solution
Given information
A randomly chosen point (X, Y)from a region S.In the region, there is an xy-plane containing all points (x,y).
Subject to \(x \ge 0,y \ge 0\)and \(4y + x \le 4\)
Define the joint p.d.f
a.
The region Sin the xy-plane is figured by,
Here the (X, Y) became the uniform distribution onS.
So, the joint distribution probability density function ofXandYis defined by,
\({{\bf{f}}_{\left( {{\bf{X,Y}}} \right)}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{1}}}{{\bf{2}}}\;{\bf{if}}\;\left( {{\bf{x,y}}} \right) \in {\bf{S}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)
Compute the probability
b.
\({S_0}\)is a subset of the region Sand has the area\(\alpha \).
We must calculate the probability that the randomly chosen point belongs to the subset\({S_0}\).
So,
Thus, the probability value is \(\frac{\alpha }{2}\).
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