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
A die is rolled thrice. What is the probability of getting a number greater than 4 in the first and the second throws, and a number less than 4 in the third throw?
Two dice are thrown simultaneously. If \( X \) denotes the number of fours, then the expectation of \( X \) will be:
Let X denote the number of hours you play during a randomly selected day. The probability that X can take values x has the following form, where c is some constant:
\[ P(X = x) = \begin{cases} 0.1, & \text{if } x = 0 \\ cx, & \text{if } x = 1 \text{ or } x = 2 \\ c(5 - x), & \text{if } x = 3 \text{ or } x = 4 \\ 0, & \text{otherwise} \end{cases} \]
Match List-I with List-II:
List-I List-II (A) c (I) 0.75 (B) P(X ≤ 2) (II) 0.3 (C) P(X = 2) (III) 0.55 (D) P(X ≥ 2) (IV) 0.15 Choose the correct answer from the options given below:
For the differential equation \( (x \log_e x) dy = (\log_e x - y) dx \):
(A) Degree of the given differential equation is 1.
(B) It is a homogeneous differential equation.
(C) Solution is \( 2y \log_e x + A = (\log_e x)^2 \), where A is an arbitrary constant.
(D) Solution is \( 2y \log_e x + A = \log_e (\log_e x) \), where A is an arbitrary constant.
Choose the correct answer from the options given below:
Which of the following is the probability of \( x \) successes in a binomial distribution with number of trials \( n \) and probability of success as \( \theta \) ( \( 0 < \theta < 1 \) ) in each trial?
