The probability distribution of a random variable $x$ is, $P(x) =\frac{k}{2^x},x= 0,1,2,3$. Then
Match List-I with List-II
|
List-I |
List-II |
|
(A) $k$ |
(I) $\frac{2}{15}$ |
|
(B) $P(x=1)$ |
(II) $\frac{1}{5}$ |
|
(C) $P(1<x<3)$ |
(III) $\frac{8}{15}$ |
|
(D) $P(x ≥2)$ |
(IV) $\frac{4}{15}$ |
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
|
List-I |
List-II |
|
(A) $k$ |
(III) $\frac{8}{15}$ |
|
(B) $P(x=1)$ |
(IV) $\frac{4}{15}$ |
|
(C) $P(1<x<3)$ |
(I) $\frac{2}{15}$ |
|
(D) $P(x ≥2)$ |
(II) $\frac{1}{5}$ |
$P(x)=\frac{k}{2^x},\;x=0,1,2,3$.
Since total probability is $1$:
$\displaystyle \frac{k}{2^0}+\frac{k}{2^1}+\frac{k}{2^2}+\frac{k}{2^3} = k\left(1+\frac12+\frac14+\frac18\right)=k\cdot\frac{15}{8}=1$
$\displaystyle k=\frac{8}{15}$
Now:
$P(x=1)=\frac{k}{2}=\frac{8}{15}\cdot\frac12=\frac{4}{15}$
$P(1
$P(x\ge 2)=P(2)+P(3)=\frac{k}{4}+\frac{k}{8} =\frac{2}{15}+\frac{1}{15}=\frac{3}{15}=\frac15$
Correct matching: A–III, B–IV, C–I, D–II