To get the standard deviation of the probability distribution, we will use the formula below
[tex]\sigma=\sqrt[]{\sum ^{}_{}(x-\bar{x})^2\times P(x)}[/tex]
where
[tex]\begin{gathered} \bar{x}=mean \\ x=\text{value } \\ P(x)=\text{probability of the value occurring} \end{gathered}[/tex]
Step 1: To begin with, we will get the mean first
[tex]\bar{x}=1\times0.12+2\times0.25+3\times0.13+4\times0.1+5\times0.1+6\times0.3=3.71[/tex]
[tex]\bar{x}=3.71[/tex]
Next, we will find
[tex]\begin{gathered} \sum ^{}_{}(x-\bar{x})^2\times p(x)=0.12(1-3.71)^2+0.25(2-3.71)^2+0.13(3-3.71)^2+0.1(4-3.71)^2+0.1(5-3.71)^2+0.3(6-3.71)^2 \\ =3.4259 \end{gathered}[/tex]
The final step will be to find the square root of the value obtained above
[tex]\sqrt[]{\sum ^{}_{}(x-\bar{x})^2\times p(x)}=\sqrt[]{3.4259}=1.851[/tex]
From the options provided, the closest answer is 1.84
Thus, the answer is 1.84
