JEE MAIN - Mathematics (2021 - 25th July Evening Shift - No. 15)
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 :
$${3 \over 8}$$ and $${1 \over 8}$$
$${3 \over 4}$$ and $${1 \over 8}$$
$${3 \over 4}$$ and $${1 \over 9}$$
$${3 \over 4}$$ and $${1 \over 16}$$
Explanation
Mean = $$\sum {{X_i}{P_i}} = \sum\limits_{r = 0}^\infty {r.{1 \over {{3^r}}} = {3 \over 4}} $$
P(X is even) $$ = {1 \over {{3^2}}} + {1 \over {{3^4}}} + ...\infty $$
$$ = {{{1 \over 9}} \over {1 - {1 \over 9}}} = {{1/9} \over {8/9}} = {1 \over 8}$$
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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