(Solved): Let \( X \sim \mathcal{N}\left(\mu, \sigma^{2}\right) \). Show that \( \operatorname{Var}(X)=\sigm ...

Let \( X \sim \mathcal{N}\left(\mu, \sigma^{2}\right) \). Show that \( \operatorname{Var}(X)=\sigma^{2} \). This problem is worth 20 points. Let \( X \sim \mathcal{N}\left(\mu, \sigma^{2}\right) \). Let \( Y=a X+b \), where \( 0 \neq a \) and \( b \) are arbitrary real numbers. Show that \( Y \) is a Normal random variable. This problem is worth 20 points.