For each of the following functions, (i) find the constant c so that f(x) is a p

Question

For each of the following functions,
(i) find the constant c so that f(x) is a p.d.f of a randomvariable X. (ii) Find the distribution function , F(x) = P(X< or =x) a) f(x)(x^3)/4,           0<x<c b) f(x)c(x),              0 <or= x <or= 4 c) f(x) = c / (x^(3/4)),  0< x<1     For each of the following functions,
(i) find the constant c so that f(x) is a p.d.f of a randomvariable X. (ii) Find the distribution function , F(x) = P(X< or =x) a) f(x)(x^3)/4,           0<x<c b) f(x)c(x),              0 <or= x <or= 4 c) f(x) = c / (x^(3/4)),  0< x<1    

Explanation / Answer

A) F(X) = int(f(x), x=0..X) = X^4/16 evaluated at x = 0..X =X^4/16 Since F(c) = 1, c^4/16 = 1 --> c = 2 B) F(X) = int(f(x), x=0..X) = 2/3 * c * X^(3/2) F(4) = 1 --> 2/3 * c * 4^(3/2) = 2/3 * c * 8 = 1 --> c =3/16 F(X) = X^(3/2) / 8 C) F(X) = int(f(x), x = 0..1) = 4cX^(1/4) F(1) = 1 --> 4c = 1 --> c = 1/4 F(X) = X^(1/4)

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