Q19 Comprehension 3 Marks 7 Jul 2024
Passage
Based on the above data, answer the given subquestions. Let X X X be a continuous random variable with the following PDF: f X ( x ) = { c x 2 ( 1 − x ) , 0 ≤ x ≤ 1 , 0 , otherwise . f_X(x)=\begin{cases}cx^2(1-x), & 0\le x\le1,\\ 0, & \text{otherwise}.\end{cases} f X ( x ) = { c x 2 ( 1 − x ) , 0 , 0 ≤ x ≤ 1 , otherwise .
Calculate the CDF of X X X . A F X ( x ) = 0 F_X(x)=0 F X ( x ) = 0 for x < 0 x<0 x < 0 ; 4 x 3 − 3 x 4 4x^3-3x^4 4 x 3 − 3 x 4 for 0 ≤ x < 1 0\le x<1 0 ≤ x < 1 ; 1 1 1 for x ≥ 1 x\ge1 x ≥ 1 B F X ( x ) = 0 F_X(x)=0 F X ( x ) = 0 for x < 0 x<0 x < 0 ; 1 12 ( x 3 3 − x 4 4 ) \frac{1}{12}\left(\frac{x^3}{3}-\frac{x^4}{4}\right) 12 1 ( 3 x 3 − 4 x 4 ) for 0 ≤ x < 1 0\le x<1 0 ≤ x < 1 ; 1 1 1 for x ≥ 1 x\ge1 x ≥ 1 C F X ( x ) = 0 F_X(x)=0 F X ( x ) = 0 for x < 0 x<0 x < 0 ; 24 x − 36 x 2 24x-36x^2 24 x − 36 x 2 for 0 ≤ x < 1 0\le x<1 0 ≤ x < 1 ; 1 1 1 for x ≥ 1 x\ge1 x ≥ 1 D F X ( x ) = 0 F_X(x)=0 F X ( x ) = 0 for x < 0 x<0 x < 0 ; x 6 − x 2 4 \frac{x}{6}-\frac{x^2}{4} 6 x − 4 x 2 for 0 ≤ x < 1 0\le x<1 0 ≤ x < 1 ; 1 1 1 for x ≥ 1 x\ge1 x ≥ 1 