# State which of the following is a probability distributions of a random variable. Give reasons for your answer.

$(i)\quad \begin{matrix} \textbf{X} & 0 &1 &2 \\ \textbf{P(X)} &0.4 &0.4 & 0.2 \end{matrix} \qquad (ii)\quad \begin{matrix} \textbf{X} &0 &1 &2 &3 &4 \\ \textbf{P(X)}&0.1 &0.5 &0.2 &-0.1 &0.3 \end{matrix}$ $(iii)\quad \begin{matrix} \textbf{Y} & -1 &0 &1 \\ \textbf{P(Y)} &0.6 &0.1 & 0.2 \end{matrix} \qquad (iv)\quad \begin{matrix} \textbf{Z} &3 &2 &1 &0 &-1 \\ \textbf{P(Z)}&0.3 &0.2 &0.4 &0.1 &0.05 \end{matrix}$

Toolbox:
• To check if a given distribution is a probability distribution of random variable, the sum of the individual probabilties should add up to 1 (i.e, $\sum P(X_i) = 1$). Also 0 $\lt$ P(X) $\leq$ 1.
(i) $\sum P(X_i)$= 0.4+0.4+0.2 = 1, and for all X, 0 $\lt$ P(X) $\leq$ 1. Hence this is a probability distribution.
(ii) P(3) = -0.1. Therefore this cannot be a probability distribution of a random variable.
(iii) $\sum P(X_i)$= 0.6+0.1+0.2 = 0.9 which is $\neq$ 1. Therefore this cannot be a probability distribution.
(iv)$\sum P(X_i) = 0.3+0.2+0.4+0.1+0.05 = 1.05$\neq\$ 1. Therefore this cannot be a probability distribution for a random variable.