\(p(x_1)\)=K\(p(x_2)=K^2\ p(x_3)=2K^2 p(x_3)\)=K
\(K+K^2+2K^2+K=1\)
\(2K+3K^2=1\)
\(3K^2+2K-1=0\)
\(3R(K+1)-1(K+1)=0\)
\((3K-1)(K+1)=0\)
3K-1=0
3K=1
K=\(\Large\frac{1}{3}\)
\(X_1\)=\(.5\)\(X_2\)=1\(X_3\)=1.5\(X_4\)=2
\(mean=\Large\;\frac{1}{3}\;\times\;.5\;+\;(\frac{1}{3})^2 +2(\frac{1}{3})^2\times1.5+\frac{1}{3}\times2\)
=\(\Large\frac{1}{6}\)+\(\Large\frac{1}{9}\)+\(\Large\frac{1}{3}\)+\(\Large\frac{2}{3}\)
=\(\Large\frac{23}{18}\)