A random variable \(X\) has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 5 } \right)\). Find
(a) \(\mathrm { E } ( \mathrm { X } )\),
(b) \(\mathrm { P } ( \mathrm { X } = 4 )\),
(c) \(P ( X > 4 )\).
A random variable \(Y\) has the distribution \(\operatorname { Geo } ( p )\), and \(q = 1 - p\).
(a) Show that \(P ( Y\) is odd \() = p + q ^ { 2 } p + q ^ { 4 } p + \ldots\).
(b) Use the formula for the sum to infinity of a geometric progression to show that
$$P ( Y \text { is odd } ) = \frac { 1 } { 1 + q }$$
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