State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

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Solution

It is known that the sum of all the probabilities in a probability distribution is one.
(i) Sum of the probabilities $= 0.4 + 0.4 + 0.2 = 1$
Therefore, the given table is a probability distribution of random variables.
(ii) It can be seen that for $X = 3, P (X) = - 0.1$
It is known that probability of any observation is not negative. Therefore, the given table is not a probability distribution of random variables.
(iii) Sum of the probabilities $= 0.6 + 0.1 + 0.2 = 0.9 \neq 1$
Therefore, the given table is not a probability distribution of random variables.
(iv) Sum of the probabilities $= 0.3 + 0.2 + 0.4 + 0.1 + 0.05 = 1.05 \neq 1$
Therefore, the given table is not a probability distribution of random variables.

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