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f(x)
=(2/π)(cosx)^2 ;-π/2≤x≤π/2
=0 ; elsewhere
E(X)
=∫(-π/2->π/2) xf(x) dx
=∫(-π/2->π/2)(2/π)x(cosx)^2 dx
= 0
E(X^2)
=∫(-π/2->π/2) x^2.f(x) dx
=∫(-π/2->π/2)(2/π) x^2. (cosx)^2 dx
=2∫(0->π/2)(2/π) x^2. (cosx)^2 dx
=(2/π)∫(0->π/2) x^2. (1+cos2x) dx
=[2/(3π)] [x^3]|(0->π/2) +(1/π)∫(0->π/2) x^2. dsin2x
=π^2/12 + (1/π) [x^2.sin2x]|(0->π/2) - (2/π)∫(0->π/2) xsin2x dx
=π^2/12 + 0 + (1/π)∫(0->π/2) xdcos2x
=π^2/12 + (1/π)[xcos2x]|(0->π/2) - (1/π)∫(0->π/2) cos2x dx
=π^2/12 - 1/2 + [1/(2π)] [ sin2x]|(0->π/2)
=π^2/12 - 1/2
D(X)
=E(X^2)-[E(X)]^2
=π^2/12 - 1/2
=(2/π)(cosx)^2 ;-π/2≤x≤π/2
=0 ; elsewhere
E(X)
=∫(-π/2->π/2) xf(x) dx
=∫(-π/2->π/2)(2/π)x(cosx)^2 dx
= 0
E(X^2)
=∫(-π/2->π/2) x^2.f(x) dx
=∫(-π/2->π/2)(2/π) x^2. (cosx)^2 dx
=2∫(0->π/2)(2/π) x^2. (cosx)^2 dx
=(2/π)∫(0->π/2) x^2. (1+cos2x) dx
=[2/(3π)] [x^3]|(0->π/2) +(1/π)∫(0->π/2) x^2. dsin2x
=π^2/12 + (1/π) [x^2.sin2x]|(0->π/2) - (2/π)∫(0->π/2) xsin2x dx
=π^2/12 + 0 + (1/π)∫(0->π/2) xdcos2x
=π^2/12 + (1/π)[xcos2x]|(0->π/2) - (1/π)∫(0->π/2) cos2x dx
=π^2/12 - 1/2 + [1/(2π)] [ sin2x]|(0->π/2)
=π^2/12 - 1/2
D(X)
=E(X^2)-[E(X)]^2
=π^2/12 - 1/2
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