State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
(i) $\begin{array}{l}X\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.2\end{array}$
(ii) $\begin{array}{l}X\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,3\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.2\,\,\,\,\,\,\,\,\, - 0.1\end{array}$
(iii) $\begin{array}{l}Y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.6\,\,\,\,\,\,\,\,\,\,\,\,\,0.1\,\,\,\,\,\,\,\,\,\,\,0.2\end{array}$
(iv) $\begin{array}{l}Z\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,0\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.3\,\,\,\,\,\,\,\,\,\,\,0.2\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\, - 0.1\end{array}$
State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
(i) $\begin{array}{l}X\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.2\end{array}$
(ii) $\begin{array}{l}X\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,3\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\,\,\,0.2\,\,\,\,\,\,\,\,\, - 0.1\end{array}$
(iii) $\begin{array}{l}Y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.6\,\,\,\,\,\,\,\,\,\,\,\,\,0.1\,\,\,\,\,\,\,\,\,\,\,0.2\end{array}$
(iv) $\begin{array}{l}Z\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,0\\P(X)\,\,\,\,\,\,\,\,\,\,\,0.3\,\,\,\,\,\,\,\,\,\,\,0.2\,\,\,\,\,\,\,\,\,\,\,0.4\,\,\,\,\,\,\,\,\, - 0.1\end{array}$
Official Solution
.: (i) This distribution is probability distribution, because sum of probabilities is one
(ii) This distribution is not a probability distribution, because probability can never be negative
(iii) This distribution is not a probability distribution, because sum of probabilities is less than one
(iv) This distribution is not a probability distribution, because sum of probabilities is more than one.
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