The probability distribution of a random variable X is:

$P(X = x)=\left\{\begin{matrix}kx^2,&for\,x=1,2,3\\2kx,&for\,x=4,5,6\\0,&\text{otherwise}\end{matrix}\right.$, where k is a constant.

Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(II), (B)-(IV), (C)-(I), (D)-(III)

The correct answer is Option (3) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)

Given

$P(X=x)=kx^2$ for $x=1,2,3$

$P(X=x)=2kx$ for $x=4,5,6$

Total probability $=1$

$k(1^2+2^2+3^2)+2k(4+5+6)=1$

$k(1+4+9)+2k(15)=1$

$14k+30k=1$

$44k=1$

$k=\frac{1}{44}$

So (A) → (II)

$P(X\ge4)=P(4)+P(5)+P(6)$

$=2k(4+5+6)=30k$

$=30\times\frac{1}{44}=\frac{15}{22}$

So (B) → (IV)

$P(X<4)=1-P(X\ge4)$

$=1-\frac{15}{22}=\frac{7}{22}$

So (C) → (I)

$E(X)=\sum xP(X=x)$

$=k(1^3+2^3+3^3)+2k(4^2+5^2+6^2)$

$=k(1+8+27)+2k(16+25+36)$

$=k(36)+2k(77)$

$=36k+154k=190k$

$=190\times\frac{1}{44}=\frac{95}{22}$

So (D) → (III)

Correct matching: A–II, B–IV, C–I, D–III