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) = {kx², for x = 1, 2, 3; 2kx, for x = 4, 5, 6; 0, otherwise} where x denotes the number of hours. Based on the above information, answer the following questions :

The probability distribution of a random variable X is: | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.1 | k | 2k | k | 0.1 | where k is some unknown constant. The probability that the random variable X takes the value 2 is:

Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is 1/5, Jaspreet's selection is 1/3 and Alia's selection is 1/4. The event of selection is independent of each other.

The random variable X has the following probability distribution where a and b are some constants : | X | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | P(X) | 0.2 | a | a | 0.2 | b | If the mean E(X) = 3, then find values of a and b and hence determine P(X ≥ 3).