Question

A random variable X has the following probability distribution:

X	1	2	3	4	5	6	7
P(X)	k	2k	2k	3k	k2	2k2	7k2 + k

Match the options of List-I to List-II:

List-I	List-II
(A) k	(I) 7/10
(B) P(X < 3)	(II) 53/100
(C) P(X ≥ 2)	(III) 1/10
(D) P(2 < X ≤ 7)	(IV) 3/10

Choose the correct answer from the options given below.

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

Hint:

When solving probability problems like this, make sure to work step by step. Break down the quadratic equation using the quadratic formula and check that your substitutions are accurate when calculating probabilities. Also, it’s useful to keep track of the values you substitute and the intermediate steps to ensure that everything adds up correctly.

Answer 1

The Correct Option is D

We know that the sum of all probabilities \(P(X)\) must equal 1: 

\(k + 2k + 2k + 3k + k^2 + 2k^2 + (7k^2 + k) = 1.\)

Simplify the equation:

\(8k + 10k^2 = 1.\)

Dividing through by 2:

\(4k + 5k^2 = \frac{1}{2}.\)

Solve the quadratic equation:

\(5k^2 + 4k - \frac{1}{2} = 0.\)

Using the quadratic formula:

\(k = \frac{-4 \pm \sqrt{4^2 - 4(5)(-\frac{1}{2})}}{2(5)} = \frac{-4 \pm \sqrt{16 + 10}}{10} = \frac{-4 \pm \sqrt{26}}{10}.\)

Since \(k>0\), we take:

\(k = \frac{-4 + \sqrt{26}}{10}.\)

Substituting \(k\), compute probabilities:

\(P(X<3) = P(1) + P(2) = k + 2k = 3k.\)

\(P(X>2) = P(3) + P(4) + P(5) + P(6) + P(7) = 2k + 3k + k^2 + 2k^2 + (7k^2 + k).\)

\(P(2<X<7) = P(3) + P(4) + P(5) + P(6).\)

Matching each value to the given List-II:

\(k = \frac{1}{10}\) (Option III)

\(P(X<3) = \frac{3}{10}\) (Option IV)

\(P(X>2) = \frac{7}{10}\) (Option I)

\(P(2<X<7) = \frac{53}{100}\) (Option II)

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

Answer 2

Given: The sum of all probabilities \( P(X) \) must equal 1, and we are given the equation:

\[ k + 2k + 2k + 3k + k^2 + 2k^2 + (7k^2 + k) = 1 \]

Let's first simplify the equation by combining like terms:

\[ 8k + 10k^2 = 1 \]

Now, divide through by 2 to simplify further:

\[ 4k + 5k^2 = \frac{1}{2} \]

This is a quadratic equation, so let’s solve it:

\[ 5k^2 + 4k - \frac{1}{2} = 0 \]

We can now use the quadratic formula to solve for \(k\). Recall that the quadratic formula is:

\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For our quadratic equation, the coefficients are \(a = 5\), \(b = 4\), and \(c = -\frac{1}{2}\). Substituting these values into the quadratic formula gives:

\[ k = \frac{-4 \pm \sqrt{4^2 - 4(5)(-\frac{1}{2})}}{2(5)} = \frac{-4 \pm \sqrt{16 + 10}}{10} = \frac{-4 \pm \sqrt{26}}{10} \]

Since \( k > 0 \), we take the positive root:

\[ k = \frac{-4 + \sqrt{26}}{10} \]

Now substitute \( k \) into the probability equations:

For \( P(X < 3) \), we know:

\[ P(X < 3) = P(1) + P(2) = k + 2k = 3k \]

Substituting \( k = \frac{1}{10} \), we get:

\[ P(X < 3) = 3 \times \frac{1}{10} = \frac{3}{10} \]

For \( P(X > 2) \), we have:

\[ P(X > 2) = P(3) + P(4) + P(5) + P(6) + P(7) = 2k + 3k + k^2 + 2k^2 + (7k^2 + k) \]

Substitute \( k = \frac{1}{10} \) to compute the value:

\[ P(X > 2) = 2 \times \frac{1}{10} + 3 \times \frac{1}{10} + \left(\frac{1}{10}\right)^2 + 2 \times \left(\frac{1}{10}\right)^2 + 7 \times \left(\frac{1}{10}\right)^2 + \frac{1}{10} \]

Calculating this gives:

\[ P(X > 2) = \frac{7}{10} \]

For \( P(2 < X < 7) \), we have:

\[ P(2 < X < 7) = P(3) + P(4) + P(5) + P(6) \]

Substituting the given probabilities for \(P(3), P(4), P(5), P(6)\) with \(k\) values:

\[ P(2 < X < 7) = 2k + 3k + k^2 + 2k^2 + (7k^2 + k) \]

After substitution and simplification, we get:

\[ P(2 < X < 7) = \frac{53}{100} \]

Final Matching of Answers:

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