(a)
"\\displaystyle\\int_{-\\infin}^{\\infin}f(x)dx=\\displaystyle\\int_{0}^{1}n(1-x)^{n-1}dx"
"=[-(1-x)^n]\\begin{matrix}\n 1 \\\\\n 0\n\\end{matrix}=-(0-1)=1, n\\geq1" Therefore "f(x)" is indeed a pdf.
(b)
"F(x)=\\displaystyle\\int_{-\\infin}^xf(t)dt" "0<x<1"
"F(x)=\\displaystyle\\int_{0}^xn(1-t)^{n-1}dt=[-(1-t)^n]\\begin{matrix}\n x \\\\\n 0\n\\end{matrix}"
"=1-(1-x)^n"
"F(x)=\\begin{cases}\n 0 &x<0\\\\\n 1-(1-x)^n & 0\\leq x<1\\\\\n1 & x\\geq 1\n\\end{cases}"
Comments
Leave a comment