Find the value of \( k \).

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 \( P(1 < X < 6) \).

Answer 1

Step 1: The sum of all probabilities must equal 1: \[P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1.\]Step 2: Input the given probability values: \[P(X = 0) = 0.1, \quad P(X = 1) = k(1), \quad P(X = 2) = k(2), \]\[P(X = 3) = k(5 - 3), \quad P(X = 4) = k(5 - 4).\]Step 3: Construct the equation: \[0.1 + k(1) + k(2) + k(2) + k(1) = 1.\]Step 4: Simplify the equation: \[0.1 + 6k = 1.\]Step 5: Calculate the value of \( k \): \[k = \frac{1 - 0.1}{6} = \frac{0.9}{6} = 0.15.\]

Find the value of \( k \).

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Solution and Explanation

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