Question

Suppose that an electronic system contains n components that function independently of each other, and suppose that these components are connected in series, as defined in Exercise 5 of Sec. 3.7. Suppose also that each component will function properly for a certain number of periods and then will fail. Finally, suppose that for i =1,...,n, the number of periods for which component i will function properly is a discrete random variable having a geometric distribution with parameter \({p_i}\). Determine the distribution of the number of periods for which the system will function properly.

Short answer

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

Step-by-step solution

Given information

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

These components are connected in series.

Calculating the distribution.

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