Question

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10 Q3. (a) Prove the Bonferroni Inequality on three events Ai, A2 and A: P(AinAnAS) 21- P(A) - P(A2)- P(As) (b) Using the results in Q3.(a), and clearly describing the events Ai, A2 and A3, construct a 100(1-a)% joint confidence intervals for estima- tion of three parameters, denoted by 0,02 and 03, say.

Answer 1

3)

a)

$P(A_1^c \cap A_2^c \cap A_3^c) = P[(A_1 \cup A_2 \cup A_3)^c]$ (By De-Moivres theorem)

$= 1 - P(A_1 \cup A_2 \cup A_3)$

$= 1 - [P(A_1)+P(A_2)+P(A_3)-P(A_1 \cap A_2)-P(A_2 \cap A_3)-P(A_1 \cap A_3)+P(A_1 \cap A_2 \cap A_3)]$

$= 1 - P(A_1)-P(A_2)-P(A_3)+P(A_1 \cap A_2)+P(A_2 \cap A_3)+P(A_1 \cap A_3)-P(A_1 \cap A_2 \cap A_3)$

We know that,

$P(A_1 \cap A_2)+P(A_2 \cap A_3)+P(A_1 \cap A_3) \ge P(A_1 \cap A_2 \cap A_3)$ (By Venn diagram)

$P(A_1 \cap A_2)+P(A_2 \cap A_3)+P(A_1 \cap A_3) - P(A_1 \cap A_2 \cap A_3) \ge 0$

Hence,

$1 - P(A_1)-P(A_2)-P(A_3)+P(A_1 \cap A_2)+P(A_2 \cap A_3)+P(A_1 \cap A_3)-P(A_1 \cap A_2 \cap A_3)$

1- P(A1) - P(A2) - P(A3)

Hence,

$P(A_1^c \cap A_2^c \cap A_3^c) \ge 1 - P(A_1)-P(A_2)-P(A_3)$