If a random variable x has the following probability distribution:

, then

Match List-I with List-II

Choose the correct answer from the options given below:

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

The correct answer is Option (4) → (A)-(III), (B)-(I), (C)-(II), (D)-(IV)

Given probability distribution:

$P(X=0) = k$, $P(X=1) = 2k$, $P(X=2) = 3k$, $P(X=3) = k^2$, $P(X=4) = 6k^2$

Sum of probabilities = 1:

$k + 2k + 3k + k^2 + 6k^2 = 6k + 7k^2 = 1$

Solve quadratic: $7k^2 + 6k - 1 = 0$

Factor: $(7k - 1)(k + 1) = 0 \Rightarrow k = 1/7$ (since probability must be positive)

Compute probabilities:

(A) $k = 1/7 \Rightarrow$ (III)

(B) $P(X<2) = P(X=0) + P(X=1) = k + 2k = 3k = 3/7 \Rightarrow$ (I)

(C) $P(X>3) = P(X=4) = 6k^2 = 6*(1/7)^2 = 6/49 \Rightarrow$ (II)

(D) $P(2 \le X \le 3) = P(X=2) + P(X=3) = 3k + k^2 = 3/7 + 1/49 = 22/49 \Rightarrow$ (IV)