GO Classes DPP | STATISTICS |VARIANCE

Answer 1

For a fair die,

$$
Y \in\{1,2,3,4,5,6\}
$$

Mean of $Y$:

$$
E[Y]=\frac{1+2+3+4+5+6}{6}=3.5
$$

Variance of $Y$:

$$
\begin{gathered}
E\left[Y^2\right]=\frac{1^2+2^2+3^2+4^2+5^2+6^2}{6}=\frac{91}{6} \\\\
\operatorname{Var}(Y)=E\left[Y^2\right]-(E[Y])^2=\frac{91}{6}-(3.5)^2=\frac{91}{6}-\frac{49}{4}=\frac{35}{12}
\end{gathered}
$$

Now, $X=2 Y+1$
 

For a linear transformation $X=a Y+b$,

$$
\operatorname{Var}(X)=a^2 \cdot \operatorname{Var}(Y)
$$

So,

$$
\operatorname{Var}(X)=4 \times \frac{35}{12}=\frac{140}{12}=\frac{35}{3}
$$