To find the value of k we will be using the very basic idea of probability.

Note: We know that the sum of the probabilities of all random variables taken from a given sample space is equal to 1.

∴ P(X=0.5) + P(X=1) + P(X=1.5) + P(X=2) = 1

∴ k + k² + 2k² + k = 1

⇒ 3k² + 2k – 1 = 0

⇒ 3k² + 3k - k – 1 = 0

⇒ 3k(k+1) – (k+1) = 0

⇒ (3k-1)(k+1) = 0

∴ k = 1/3 or k = -1

∵ k represents probability of an event. Hence 0≤P(X)≤1

∴ k = 1/3

Mean of any probability distribution is given by- Mean = ∑x_ip_i

Now we have,

X: 0.5 1 1.5 2

P(X): 1/3 1/9 2/9 1/3

∴ first we need to find the product i.e. p_ix_i and add them to get mean.

∴ Mean = 0.5 x (1/3) + 1 x (1/9) + 1.5 x (2/9) +2 x (1/3)