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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

Expert verified

(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

01

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.

02

Plotting the graph

(a)

The graph for\({\rm{f(x)}}\)is plotted as 鈥�

Therefore, the graph is obtained.

03

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 \({\rm{0}}{\rm{.5}}\).

04

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 \({\rm{0}}{\rm{.6875}}\).

05

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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Most popular questions from this chapter

The article 鈥淢odelling Sediment and Water Column Interactions for Hydrophobic Pollutants鈥� (Water Research, \({\rm{1984: 1169 - 1174}}\)) suggests the uniform distribution on the interval \({\rm{(7}}{\rm{.5,20)}}\)as a model for depth (cm) of the bioturbation layer in sediment in a certain region. a. What are the mean and variance of depth? b. What is the cdf of depth? c. What is the probability that observed depth is at most \({\rm{10}}\)? Between \({\rm{10}}\) and \({\rm{15}}\)? d. What is the probability that the observed depth is within \({\rm{1}}\) standard deviation of the mean value? Within \({\rm{2 }}\) standard deviations?

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