Let X be a discrete random variable whose probability distribution is defined as follows:


where k is a constant. Calculate


(i) the value of k (ii) E (X) (iii) Standard deviation of X.

Given:


Thus, we have the probability distribution of X is



(i) the value of k


We know that,


Sum of the probabilities = 1



∴ 2k + 3k + 4k + 5k + 10k + 12k + 14k = 1


⇒ 50k = 1



⇒ k = 0.02


(ii) To find: E(X)


The probability distribution of X is:



Therefore,


μ = E(X)



∴ E(X) = 2k + 6k + 12k + 20k + 50k + 72k + 98k + 0


= 260k




= 5.2 …(i)


(iii) To find: Standard deviation of X



We know that,


Var(X) = E(X2) – [E(X)]2


= ΣX2P(X) – [Σ{XP(X)}]2


= [2k + 12k + 36k + 80k + 250k + 432k + 686k +0] – [5.2]2 = 1498k – 27.04



= 29.96 – 27.04


= 2.92


We know that,


standard deviation of X = √Var(X) = √2.92 = 1.7088


≅ 1.7

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