Let X be a random variable such that the probability function of a distribution is given by P ( X =0)=1/2, P ( X = j )=1/(3^j)( j =1,2,3, ….., ∞)…

Let $\displaystyle X$ be a random variable such that the probability function of a distribution is given by $\displaystyle \mathrm{P}(\mathrm{X}=0)=\frac{1}{2}, \mathrm{P}(\mathrm{X}=\mathrm{j})=\frac{1}{3^{\mathrm{j}}}(\mathrm{j}=1,2,3, \ldots . ., \infty)$ Then the mean of the distribution and $\displaystyle \mathrm{P}(\mathrm{X}$ is positive and even) respectively are :