(a) Probability density function of a random veriable X is given by
f(x)=kx(2−x) , 0<x<2
=0, e.w
We know that for p.d.f ,
∫−∞∞f(x)dx=1
⟹∫02kx(2−x)dx=1
⟹k.∫02(2x−x2)dx=1
⟹k.[2.2x2−3x3]02=1
⟹k.[4−38]=1
⟹k=43
(b) The distribution function of the continuous random variable X having probability density function f(x) is given by ,
FX(x)=P(X≤x)=∫−∞xf(u)du
Therefore for the given p.d.f , the distribution function FX(x)=∫−∞xf(u)du
=∫0x43u(2−u)du
=43∫0x(2u−u2)du
=43[u2−3u3]0x
=41[3x2−x3]
(c) P(X>3)=∫3∞f(x)dx=0 [ as f(x)=0 when 3<x<∞ ]
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