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A probability distribution indicates the likelihood of a random variable to take a particular value within a given range.In a probability distribution, the percentages of probabilities are mainly represented as decimals.

With the help of intuition, the mean can be found by finding the average of the probabilities as shown below:

$$\begin{gathered} \mathbf{\text{²Ñ±ð²¹²Ô f´Ç°ù x=}}\frac{\mathbf{\text{0.3+0.4+0.3}}}{\mathbf{\text{3}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.33}} \\ \mathbf{\text{²Ñ±ð²¹²Ô f´Ç°ù }}\mathbf{y}\mathbf{\text{=}}\frac{\mathbf{\text{0.1+0.8+0.1}}}{\mathbf{\text{3}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.33}} \end{gathered}$$

Therefore, the mean of the distributions is 0.33 for both.

Finding a variance determines the variation in the values of a dataset around the mean of the values. The researcher often uses it to determine the consistency of a particular dataset.

The second distribution appears to be more variable compared to the first one. It is because

$\begin{array}{c}\mathbf{\text{³Õ²¹°ù¾±²¹²Ô³¦±ð o´Ú x d¾±²õ³Ù°ù¾±²ú³Ü³Ù¾±´Ç²Ô=}}{\mathbf{(}\mathbf{\text{0.3}}\mathbf{−}\mathbf{\text{0.33}}\mathbf{)}}^{\mathbf{\text{2}}}\mathbf{\text{+}}{\mathbf{(}\mathbf{\text{0.4}}\mathbf{−}\mathbf{\text{0.33}}\mathbf{)}}^{\mathbf{\text{2}}}\mathbf{\text{+}}{\mathbf{(}\mathbf{\text{0.3}}\mathbf{−}\mathbf{\text{0.33}}\mathbf{)}}^{\mathbf{\text{2}}}\\ \mathbf{\text{=0.0067}}\\ \mathbf{\text{³Õ²¹°ù¾±²¹²Ô³¦±ð o´Ú }}\mathbf{y}\mathbf{\text{ d¾±²õ³Ù°ù¾±²ú³Ü³Ù¾±´Ç²Ô=}}{\mathbf{(}\mathbf{\text{0.1}}\mathbf{−}\mathbf{\text{0.33}}\mathbf{)}}^{\mathbf{\text{2}}}\mathbf{\text{+}}{\mathbf{(}\mathbf{\text{0.8}}\mathbf{−}\mathbf{\text{0.33}}\mathbf{)}}^{\mathbf{\text{2}}}\mathbf{\text{+}}{\mathbf{(}\mathbf{\text{0.1}}\mathbf{−}\mathbf{\text{0.33}}\mathbf{)}}^{\mathbf{\text{2}}}\\ \mathbf{\text{=0.3267}}\end{array}$

From the variances in the probabilities, it can be found that the y distribution has a greater variance.

The means of x and y distributions are shown below:

$\begin{array}{c}\mathbf{}\mathbf{}\mathbf{}{\mathbf{μ}}_{\mathbf{X}}\mathbf{\text{=0}}\mathbf{×}\mathbf{\text{0.3+1}}\mathbf{×}\mathbf{\text{0.4+2}}\mathbf{×}\mathbf{\text{0.3}}\\ \mathbf{\text{=1}}\\ {\mathbf{μ}}_{\mathbf{y}}\mathbf{\text{=0}}\mathbf{×}\mathbf{\text{0.1+1}}\mathbf{×}\mathbf{\text{0.8+2}}\mathbf{×}\mathbf{\text{0.1}}\\ \mathbf{\text{=1}}\end{array}$

In comparing part a, it can be found that in part c, the means of y and x distribution are equal, as observed in part a.

The variances of x and y distributions are shown below:

$$\begin{gathered} \mathbf{Variance}\text{ }\mathbf{of}\text{ }\mathbf{x}\text{ }\mathbf{distribution}\mathbf{=}{(\mathbf{0}−\mathbf{1})}^2\mathbf{×}\mathbf{0}\mathbf{.}\mathbf{3}\mathbf{+}{(\mathbf{1}−\mathbf{1})}^2\mathbf{×}\mathbf{0}\mathbf{.}\mathbf{4}\mathbf{+}{(\mathbf{2}−\mathbf{1})}^2\mathbf{×}\mathbf{0}\mathbf{.}\mathbf{3} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9} \\ \mathbf{Variance}\mathbf{\text{ }}\mathbf{of}\mathbf{\text{ }}\mathbf{y}\mathbf{\text{ }}\mathbf{distribution}\mathbf{=}{\mathbf{(}\mathbf{0}\mathbf{−}\mathbf{1}\mathbf{)}}^2\mathbf{×}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{+}{\mathbf{(}\mathbf{1}\mathbf{−}\mathbf{1}\mathbf{)}}^2\mathbf{×}\mathbf{0}\mathbf{.}\mathbf{8}\mathbf{+}{\mathbf{(}\mathbf{2}\mathbf{−}\mathbf{1}\mathbf{)}}^2\mathbf{×}\mathbf{0}\mathbf{.}\mathbf{1} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2} \end{gathered}$$

In comparing to part b, it can be found that in part b, the variance of y distribution appeared to be greater than x, but in part c, it is the opposite.