The probability distribution function of a random variable $x$ is given by

$p(x)=A(1+2x{)}^2,x=-2,0,1,2.$

a$-$ Find the value of $A.$

b$-$ Find the expected value of $x.$

c$-$ Find the standard deviation of $x.$

d$-$ Determine the cumulative distribution function of $x,$ denoted by $F(t),$ where $F(t)=P(xt)$

for all real numbers $t.$ Graph the function $F(t).$

e$-$ Compute the moment generating function of $x,$ denoted by $M_x(t),$ where $M_x(t)=E(e^{tx}).$

f$-$ Let $Y=3x^2+1,$ find $E(Y)$ and Var$(Y).$