Let $X$ be a discrete random variable and let $Y = 0.5 X + 3$. Given PMF of $X$, find PMF of $Y, E [Y ]$ and $Var[Y]$.

Question

Question : Let $X$ be a discrete random variable and let $Y = 0.5 X + 3$

1. Assume that the PMF of $X$ is given by, where $k$ is some suitable constant. Determine the value of $k$.

$\frac{1}{40}$

2. Find $E[X]$ and $Var[X]$.

$E[X]=0, Var[X]=0$

3. Given that $X$ has the PMF found above, find the PMF of $Y$ .

4. Without explicit calculation from the PMF of $Y$, find $E [Y ]$ and $Var[Y]$.

I have found answer for 1. and 2. Please help me in remaining parts.