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% 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: 
1. Express the probability distribution given above in the form of a probability distribution table. 
2. Find the value of \( k \). 
3. (a) Find the mean number of hours spent by the student. (b) Find \( P(1 < X < 6) \).

Hint:

Use the given probability function to calculate probabilities for each value of \( x \) and organize them in a table.

Answer 1

(i) Probability Distribution Table: Using the given \( P(X = x) \), compute the probabilities for \( x = 1, 2, 3, 4, 5, 6 \): \[ \begin{array}{|c|c|} \hline x & P(X = x) \\ \hline 1 & k(1^2) = k \\ 2 & k(2^2) = 4k \\ 3 & k(3^2) = 9k \\ 4 & 2k(4) = 8k \\ 5 & 2k(5) = 10k \\ 6 & 2k(6) = 12k \\ \hline \end{array} \]