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Let X be a continuous random variable with p.d.f. f(x) = kx(1-x), 0b).. find k and determine anumber b such that `P(X le b) ` = `P (X ge b)`

Answer» `kint_0^1(x(1-x))dx=1`
`k[int_0^1xdx-int_0^1x^2dx]=1`
`k[x^2/2--x^3/3]_0^1`
`k[1/2-1/3]=1`
`k=6`
`P(0,b)=P(b,1)`
`6int_0^b(x-x^2)dx=1/2`
`6[x^2/2-x^3/3]_0^b=1/2`
`b^2/2-b^3/3=1/12`
`4b^3-6b^2+1=0`
`When b=1/2`
`4/8-6/4+1=0`
`4b^3-6b^2+1=(2b-1)(2b^2-2b-1)`
`2b-1=0`
`b=1/2`
or
`2b^2-2b-1=0`
`b=(2pmsqrt12)/4`
`=1/2pmsqrt3/2`
Which is not possible so,
`1/2+sqrt3/2>1,1/2-sqrt3/2<0`.


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