If the random variable \( X \) has the following distribution: Match List-I with List-II: Choose the correct answer from the options given below: X 0 1 2 otherwise P(X) k 2k 3k 0 List-I List-II (A) k (I) \(\frac{5}{6}\) (B) P(X < 2) (II) \(\frac{4}{3}\) (C) E(X) (III) \(\frac{1}{2}\) (D) P(1 ≤ X ≤ 2) (IV) \(\frac{1}{6}\)
(A) - (IV), (B) - (III), (C) - (II), (D) - (I)
To solve the given problem, we need to match the elements of List-I with those in List-II. Let's begin by analyzing the distribution and calculating each value step-by-step.
Given, the probability distribution of the random variable \( X \) is as follows:
| X | 0 | 1 | 2 | otherwise |
|---|---|---|---|---|
| P(X) | k | 2k | 3k | 0 |
Step 1: Find k
Total probability = 1
k + 2k + 3k = 6k = 1
k = 1/6
Step 2: Find P(X < 2)
P(X < 2) = P(0) + P(1)
= k + 2k = 3k
= 3 × (1/6) = 1/2
Step 3: Find E(X)
E(X) = Σ x·P(x)
= 0·k + 1·(2k) + 2·(3k)
= 0 + 2k + 6k
= 8k
= 8 × (1/6) = 4/3
Step 4: Find P(1 ≤ X ≤ 2)
P(1 ≤ X ≤ 2) = P(1) + P(2)
= 2k + 3k = 5k
= 5 × (1/6) = 5/6
Correct option: (4)
Important Questions from Probability
Rakesh is 17th from the right and Ankit is 15th from the left in a line of students. If they interchange their places, the position of Ankit becomes 19th from the left. How many students are there in the line?
What comes in place of the question mark (?) in the series given below?
B2D, C3F, E5J, G7N, ?, M13Z
If 1st January, 2001 was a Monday, what was the day on 26th January, 2003?
From the given options, at what angle are the hands of a clock inclined at 10 minutes to 2 (Smaller angle)?
In the given analogy, choose the number which will replace the question mark (?).
WSH : 5 : : KMJ : ?
