If $X$ follows the distribution $\mathcal { R }$ (where $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$), specify the distribution of the random variable $\frac { 1 } { 2 } ( X + 1 )$.
If $X$ follows the distribution $\mathcal { R }$ (where $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$), specify the distribution of the random variable $\frac { 1 } { 2 } ( X + 1 )$.