Introduction to Probability and Statistics for Engineers and Scientists

5.1The Bernoulli and Binomial Random Variables 145

whereas the corresponding probability for a 3-component system is

(

3

2

)

p^2 (1−p)+p^3

Hence, the 5-component system is better if

10 p^3 (1−p)^2 + 5 p^4 (1−p)+p^5 ≥ 3 p^2 (1−p)+p^3

which reduces to

3(p−1)^2 (2p−1)≥ 0

or

p≥

1

2

(b)In general, a system with 2k+1 components will be better than one with 2k− 1

components if (and only if)p≥^12. To prove this, consider a system of 2k+ 1

components and letXdenote the number of the first 2k−1 that function. Then

P 2 k+ 1 (effective)=P{X≥k+ 1 }+P{X=k}(1−(1−p)^2 )+P{X=k− 1 }p^2

which follows since the 2k+1 component system will be effective if either

(1) X≥k+1;

(2) X=kand at least one of the remaining 2 components function; or

(3) X=k−1 and both of the next 2 function.

Because

P 2 k− 1 (effective)=P{X≥k}

=P{X=k}+P{X≥k+ 1 }

we obtain that

P 2 k+ 1 (effective)−P 2 k− 1 (effective)

=P{X=k− 1 }p^2 −(1−p)^2 P{X=k}

=

(

2 k− 1

k− 1

)

pk−^1 (1−p)kp^2 −(1−p)^2

(

2 k− 1

k

)

pk(1−p)k−^1

=

(

2 k− 1

k

)

pk(1−p)k[p−(1−p)] since

(

2 k− 1

k− 1

)

=

(

2 k− 1

k

)

≥ 0 ⇔p≥

1

2

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