The probability distribution of a random variable \(X\) is given below. If \(V\) is the variance of \(X\), then \(k+V=\) \[ \begin{array}{|c|c|c|c|c|} \hline X=x_i & 2 & 3 & 5 & 7 \\ \hline P(X=x_i) & 4k & 3k & 2k & k \\ \hline \end{array} \]

Question

Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% of learners were self-taught using internet resources and upskilled themselves.

A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below:

\[ P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3, \\ 2kx, & \text{for } x = 4, 5, 6, \\ 0, & \text{otherwise.} \end{cases} \]

where \( x \) denotes the number of hours. Based on the above information, answer the following questions:
(i) Express the probability distribution given above in the form of a probability distribution table.
(ii) Find the value of \( k \).
(iii)(a) Find the mean number of hours spent by the student.
(iii)(b) Find \( P(1 < X < 6) \).

The probability distribution of a random variable \(X\) is given below. If \(V\) is the variance of \(X\), then \(k+V=\) \[ \begin{array}{|c|c|c|c|c|} \hline X=x_i & 2 & 3 & 5 & 7 \\ \hline P(X=x_i) & 4k & 3k & 2k & k \\ \hline \end{array} \]

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The Correct Option is A

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