91Ó°ÊÓ

Probability refers to the likelihood of a random event's outcome. This word refers to determining the likelihood of a given occurrence occurring.

(a)

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

Therefore, the graph is obtained.

(b)

\({\rm{P(X > 0)}}\)for the given density function can be given as –

\(\begin{aligned} P(X > 0) &= \int_{\rm{0}}^{\rm{2}} {\left( {{\rm{0}}{\rm{.09375}}\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)} \right)} \cdot {\rm{dx}}\\ &= \int_{\rm{0}}^{\rm{2}} {{\rm{(0}}{\rm{.09375)(4)}}} \cdot {\rm{dx - }}\int_{\rm{0}}^{\rm{2}} {{\rm{(0}}{\rm{.09375)}}} \left( {{{\rm{x}}^{\rm{2}}}} \right) \cdot {\rm{dx}}\\ &= 0 {\rm{.375}}\int_{\rm{0}}^{\rm{2}} {\rm{d}} {\rm{x - 0}}{\rm{.09375}}\int_{\rm{0}}^{\rm{2}} {{{\rm{x}}^{\rm{2}}}} \cdot {\rm{dx}}\\ &= 0 {\rm{.375(x)}}_{\rm{0}}^{\rm{2}}{\rm{ - 0}}{\rm{.09375}}\left( {\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right)_{\rm{0}}^{\rm{2}}\\ &= 0 {\rm{.375(2 - 0) - 0}}{\rm{.09375}}\left( {\frac{{{{\rm{2}}^{\rm{3}}}{\rm{ - }}{{\rm{0}}^{\rm{3}}}}}{{\rm{3}}}} \right)\\ &= 0 {\rm{.75 - 0}}{\rm{.09375 \times }}\frac{{\rm{8}}}{{\rm{3}}}\\ &= 0 {\rm{.75 - 0}}{\rm{.25 = 0}}{\rm{.5}}\end{aligned}\)

Therefore, the value is obtained as \({\rm{0}}{\rm{.5}}\).

(c)

\({\rm{P( - 1 < X < 1)}}\)for the given density function can be given as –

\(\begin{aligned} P( - 1 < X < 1) &= \int_{{\rm{ - 1}}}^{\rm{1}} {\left( {{\rm{0}}{\rm{.09375}}\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)} \right)} \cdot {\rm{dx}}\\ &= \int_{{\rm{ - 1}}}^{\rm{1}} {{\rm{(0}}{\rm{.09375)(4)}}} \cdot {\rm{dx - }}\int_{{\rm{ - 1}}}^{\rm{1}} {{\rm{(0}}{\rm{.09375)}}} \left( {{{\rm{x}}^{\rm{2}}}} \right) \cdot {\rm{dx}}\\ &= 0 {\rm{.375}}\int_{{\rm{ - 1}}}^{\rm{1}} {\rm{d}} {\rm{x - 0}}{\rm{.09375}}\int_{{\rm{ - 1}}}^{\rm{1}} {{{\rm{x}}^{\rm{2}}}} {\rm{ \times dx}}\\ &= 0 {\rm{.375(x)}}_{{\rm{ - 1}}}^{\rm{1}}{\rm{ - 0}}{\rm{.09375}}\left( {\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right)_{{\rm{ - 1}}}^{\rm{1}}\\ &= 0 {\rm{.375(1 - ( - 1)) - 0}}{\rm{.09375}}\left( {\frac{{{{\rm{1}}^{\rm{3}}}{\rm{ - ( - 1}}{{\rm{)}}^{\rm{3}}}}}{{\rm{3}}}} \right)\\ &= 0 {\rm{.75 - 0}}{\rm{.09375 \times }}\frac{{\rm{2}}}{{\rm{3}}} &= 0 {\rm{.6875}}\end{aligned}\)

Therefore, the value is obtained as \({\rm{0}}{\rm{.6875}}\).

(d)

\({\rm{P(X < - 0}}{\rm{.5 or X > 0}}{\rm{.5)}}\) for the given density function can be given as –

\({\rm{P(X < - 0}}{\rm{.5 or X > 0}}{\rm{.5) = 1 - P( - 0}}{\rm{.5}} \le {\rm{X}} \le {\rm{0}}{\rm{.5)}}\)

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 –

\(\begin{aligned}{\rm{P(X < - 0}}{\rm{.5 or X > 0}} .5) &= 1 - P( - 0{\rm{.5}} \le {\rm{X}} \le {\rm{0}}{\rm{.5)}}\\ &= 1 - 0 {\rm{.3672}}\\ & = 0{\rm{.6328}}\end{aligned}\)

Therefore, the value is obtained as \({\rm{0}}{\rm{.6328}}\).