continuous random variable x has probability density function f(x)= k(x-1)^2, 0<x<2 0 otherwise find the value of constant k hence compute p(x<1.5)

Question

question is on continuous random variable and pdf

Answer 1

∫f(x)dx=1 for 0<x<2 (I presume) (sum total of all probabilities must be equal to 1),

k∫₀²(x-1)²dx=k∫₀²(x²-2x+1)dx=k(x³/3-x²+x)₀²=1,

k(8/3-4+2)=1,

2k/3=1, k=3/2. f(x)=(3/2)(x-1)².

p(x<1.5)=(3/2)(x³/3-x²+x)₀^1.5=1.5(1.125-2.25+1.5)=0.5625 or 56.25%.