Let X be a random variable such that the probability function of a distribution is given b…

Question

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

Accepted Answer

Mean = $$\sum {{X_i}{P_i}}  = \sum\limits_{r = 0}^\infty  {r.{1 \over {{3^r}}} = {3 \over 4}} $$

P(X is even) $$ = {1 \over {{3^2}}} + {1 \over {{3^4}}} + ...\infty $$

$$ = {{{1 \over 9}} \over {1 - {1 \over 9}}} = {{1/9} \over {8/9}} = {1 \over 8}$$

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