91Ó°ÊÓ

An Electronic system contain n components that function independently of each other.

These components are connected in series.

Since the components are connected in series, the system will function properly only as long as every component function properly.

Let\({X_i}\)be the number of periods that component ifunctions properly.

For\(i = 1,2,...n\)let\(X\)denote period that system functions properly. then for any nonnegative integer x

\(\begin{aligned}{c}{\rm P}\left( {X \ge x} \right) &= {\rm P}\left( {{X_1} \ge x,....,{X_n} \ge x.} \right)\\ &= {\rm P}\left( {{X_1} \ge x} \right)....{\rm P}\left( {{X_n} \ge x} \right)\end{aligned}\)

because the n components are independent

\({\rm P}\left( {{X_i} \ge x} \right) = {\left( {1 - {p_i}} \right)^x}\)

Therefore\({\rm P}\left( {X \ge x} \right) = \prod\limits_{i = 1}^n {{{\left( {1 - {p_i}} \right)}^x}} \)

It follows that

\(\begin{aligned}{c}{\rm P}\left( {X = x} \right) &= {\rm P}\left( {X \ge x} \right) - {\rm P}\left( {X \ge x + 1} \right)\\ &= \prod\limits_{i = 1}^n {{{\left( {1 - {p_i}} \right)}^x}} - \prod\limits_{i = 1}^n {{{\left( {1 - {p_i}} \right)}^{x + 1}}} \end{aligned}\)

Therefore

\({\rm P}\left( {X = x} \right) = \left( {1 - \prod\limits_{i = 1}^n {\left( {1 - {p_i}} \right)} } \right){\left( {\prod\limits_{i = 1}^n {\left( {1 - {p_i}} \right)} } \right)^x}\)

Hence it can be seen that this is the pf of the geometric distribution with

\(p = 1 - \prod\limits_{i = 1}^n {\left( {1 - {p_i}} \right)} \)

Therefore number of period for which the system will function properly will follow geometric distribution with \(p = 1 - \prod\limits_{i = 1}^n {\left( {1 - {p_i}} \right)} \)