A.
Function f(x) is a probability density function in range a to b if
∫abf(x)dx=1
Then:
∫01(6x(1−x))dx=(3x2−2x3)∣01=3−2=1
C.
E(X)=∫abxf(x)dx=∫01x(6x(1−x))dx=(2x3−1.5x4)∣01=2−1.5=0.5
B.
P(x<b)=∫0bf(x)dx
P(x>b)=1−∫0bf(x)dx
Then:
∫0bf(x)dx=1−∫0bf(x)dx
2∫0bf(x)dx=2∫0b(6x(1−x))dx=2(3x2−2x3)∣0b=6b2−4b3=1
b=0.5
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