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Magazine Chapter 4: Q3E (page 146)
The error involved in making a certain measurement is a continuous rv \({\rm{X}}\) with pdf
\({\rm{f(x) = \{ }}\begin{array}{*{20}{c}}{{\rm{.09375(4 - }}{{\rm{x}}^2})}&{{\rm{ - 2}} \le {\rm{x}} \le {\rm{2}}}\\{\rm{0}}&{{\rm{otherwise}}}\end{array}\)
a. Sketch the graph of \({\rm{f(x)}}\).
b. Compute \({\rm{P(X > 0)}}\).
c. Compute \({\rm{P( - 1 < X < 1)}}\).
d. Compute \({\rm{P(X < - }}{\rm{.5 or X > }}{\rm{.5)}}\).
Short Answer
(a) The graph for\({\rm{f(x)}}\)is -
(b) On computing \({\rm{P(X > 0)}}\) the value obtained is \({\rm{0}}{\rm{.5}}\).
(c) On computing\({\rm{P( - 1 < X < 1)}}\)the value obtained is\({\rm{0}}{\rm{.6875}}\).
(d) On computing \({\rm{P(X < - 0}}{\rm{.5 or X > 0}}{\rm{.5)}}\) the value obtained is \({\rm{0}}{\rm{.6328}}\).
Step by step solution
Concept Introduction
Probability refers to the likelihood of a random event's outcome. This word refers to determining the likelihood of a given occurrence occurring.
Plotting the graph
(a)
The graph for\({\rm{f(x)}}\)is plotted as 鈥�
Therefore, the graph is obtained.
Step 3: Computing \({\rm{P(X > 0)}}\)
(b)
\({\rm{P(X > 0)}}\)for the given density function can be given as 鈥�
\(\begin{aligned} P(X > 0) &= \int_{\rm{0}}^{\rm{2}} {\left( {{\rm{0}}{\rm{.09375}}\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)} \right)} \cdot {\rm{dx}}\\ &= \int_{\rm{0}}^{\rm{2}} {{\rm{(0}}{\rm{.09375)(4)}}} \cdot {\rm{dx - }}\int_{\rm{0}}^{\rm{2}} {{\rm{(0}}{\rm{.09375)}}} \left( {{{\rm{x}}^{\rm{2}}}} \right) \cdot {\rm{dx}}\\ &= 0 {\rm{.375}}\int_{\rm{0}}^{\rm{2}} {\rm{d}} {\rm{x - 0}}{\rm{.09375}}\int_{\rm{0}}^{\rm{2}} {{{\rm{x}}^{\rm{2}}}} \cdot {\rm{dx}}\\ &= 0 {\rm{.375(x)}}_{\rm{0}}^{\rm{2}}{\rm{ - 0}}{\rm{.09375}}\left( {\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right)_{\rm{0}}^{\rm{2}}\\ &= 0 {\rm{.375(2 - 0) - 0}}{\rm{.09375}}\left( {\frac{{{{\rm{2}}^{\rm{3}}}{\rm{ - }}{{\rm{0}}^{\rm{3}}}}}{{\rm{3}}}} \right)\\ &= 0 {\rm{.75 - 0}}{\rm{.09375 \times }}\frac{{\rm{8}}}{{\rm{3}}}\\ &= 0 {\rm{.75 - 0}}{\rm{.25 = 0}}{\rm{.5}}\end{aligned}\)
Therefore, the value is obtained as \({\rm{0}}{\rm{.5}}\).
Step 4: Computing \({\rm{P( - 1 < X < 1)}}\)
(c)
\({\rm{P( - 1 < X < 1)}}\)for the given density function can be given as 鈥�
\(\begin{aligned} P( - 1 < X < 1) &= \int_{{\rm{ - 1}}}^{\rm{1}} {\left( {{\rm{0}}{\rm{.09375}}\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)} \right)} \cdot {\rm{dx}}\\ &= \int_{{\rm{ - 1}}}^{\rm{1}} {{\rm{(0}}{\rm{.09375)(4)}}} \cdot {\rm{dx - }}\int_{{\rm{ - 1}}}^{\rm{1}} {{\rm{(0}}{\rm{.09375)}}} \left( {{{\rm{x}}^{\rm{2}}}} \right) \cdot {\rm{dx}}\\ &= 0 {\rm{.375}}\int_{{\rm{ - 1}}}^{\rm{1}} {\rm{d}} {\rm{x - 0}}{\rm{.09375}}\int_{{\rm{ - 1}}}^{\rm{1}} {{{\rm{x}}^{\rm{2}}}} {\rm{ \times dx}}\\ &= 0 {\rm{.375(x)}}_{{\rm{ - 1}}}^{\rm{1}}{\rm{ - 0}}{\rm{.09375}}\left( {\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right)_{{\rm{ - 1}}}^{\rm{1}}\\ &= 0 {\rm{.375(1 - ( - 1)) - 0}}{\rm{.09375}}\left( {\frac{{{{\rm{1}}^{\rm{3}}}{\rm{ - ( - 1}}{{\rm{)}}^{\rm{3}}}}}{{\rm{3}}}} \right)\\ &= 0 {\rm{.75 - 0}}{\rm{.09375 \times }}\frac{{\rm{2}}}{{\rm{3}}} &= 0 {\rm{.6875}}\end{aligned}\)
Therefore, the value is obtained as \({\rm{0}}{\rm{.6875}}\).
Step 5: Computing \({\rm{P(X < - 0}}{\rm{.5 or X > 0}}{\rm{.5)}}\)
(d)
\({\rm{P(X < - 0}}{\rm{.5 or X > 0}}{\rm{.5)}}\) for the given density function can be given as 鈥�
\({\rm{P(X < - 0}}{\rm{.5 or X > 0}}{\rm{.5) = 1 - P( - 0}}{\rm{.5}} \le {\rm{X}} \le {\rm{0}}{\rm{.5)}}\)
First calculate \({\rm{P( - 0}}{\rm{.5}} \le {\rm{X}} \le {\rm{0}}{\rm{.5)}}\).
\(\begin{aligned}{\rm{P( - 0}}{\rm{.5}} \le {\rm{X}} \le {\rm{0}}5) &= \int_{{\rm{ - 0}}{\rm{.5}}}^{{\rm{0}}{\rm{.5}}} {\left( {{\rm{0}}{\rm{.09375}}\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)} \right)} \cdot {\rm{dx}}\\ &= \int_{{\rm{ - 0}}{\rm{.5}}}^{{\rm{0}}{\rm{.5}}} {{\rm{(0}}{\rm{.09375)(4)}}} \cdot {\rm{dx - }}\int_{{\rm{ - 0}}{\rm{.5}}}^{{\rm{0}}{\rm{.5}}} {{\rm{(0}}{\rm{.09375)}}} \left( {{{\rm{x}}^{\rm{2}}}} \right) \cdot {\rm{dx}}\\ &= 0{\rm{.375}}\int_{{\rm{ - 0}}{\rm{.5}}}^{{\rm{0}}{\rm{.5}}} {\rm{d}} {\rm{x - 0}}{\rm{.09375}}\int_{{\rm{ - 0}}{\rm{.5}}}^{{\rm{0}}{\rm{.5}}} {{{\rm{x}}^{\rm{2}}}} \cdot {\rm{dx}}\\ &= 0 {\rm{.375(x)}}_{{\rm{ - 0}}{\rm{.5}}}^{{\rm{0}}{\rm{.5}}}{\rm{ - 0}}{\rm{.09375}}\left( {\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right)_{{\rm{ - 0}}{\rm{.5}}}^{{\rm{0}}{\rm{.5}}}\\ &= 0{\rm{.375(0}}{\rm{.5 - ( - 0}}{\rm{.5)) - 0}}{\rm{.09375}}\left( {\frac{{{\rm{0}}{\rm{.}}{{\rm{5}}^{\rm{3}}}{\rm{ - ( - 0}}{\rm{.5}}{{\rm{)}}^{\rm{3}}}}}{{\rm{3}}}} \right)\\ &= 0 {\rm{.375 - 0}}{\rm{.09375 \times }}\frac{{{\rm{0}}{\rm{.25}}}}{{\rm{3}}}\\ &= 0 {\rm{.3672}}\end{aligned}\)
Hence, it is obtained 鈥�
\(\begin{aligned}{\rm{P(X < - 0}}{\rm{.5 or X > 0}} .5) &= 1 - P( - 0{\rm{.5}} \le {\rm{X}} \le {\rm{0}}{\rm{.5)}}\\ &= 1 - 0 {\rm{.3672}}\\ & = 0{\rm{.6328}}\end{aligned}\)
Therefore, the value is obtained as \({\rm{0}}{\rm{.6328}}\).
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