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4. Prove the Bonferroni inequalities, which state that for any events {A₁, A2,..., An} CF and any integer k € {1,2,..., n}, (≤Σ=1(-1)²-¹ Σsc{1,2,...} PjesA;), if k is odd, P(U-1A₂) |≥t=1(-1)-1 Esc{1,2,...} (jesA;), if k is even. |S|=l |S|=l Do this by expanding 1 {U;_14;} = 1 – II;_1(1 − 1 {A;}), bound- =1 ing the product (as a function) from above and below, and taking expectation. (You may use without proof that if F ≤ G as random variables then EF ≤ EG and E1 {A} = P(A). We have shown this for expectation on countable probability spaces, and it remains true in general.) The special case of k = 1 is known as the union bound. When k = n, this is an equality which is known as the principle of inclusion-exclusion.

4. Prove the Bonferroni inequalities, which state that for any events {A₁, A2,..., An} CF and any integer k € {1,2,..., n}, (≤Σ=1(-1)²-¹ Σsc{1,2,...} PjesA;), if k is odd, P(U-1A₂) |≥t=1(-1)-1 Esc{1,2,...} (jesA;), if k is even. |S|=l |S|=l Do this by expanding 1 {U;_14;} = 1 – II;_1(1 − 1 {A;}), bound- =1 ing the product (as a function) from above and below, and taking expectation. (You may use without proof that if F ≤ G as random variables then EF ≤ EG and E1 {A} = P(A). We have shown this for expectation on countable probability spaces, and it remains true in general.) The special case of k = 1 is known as the union bound. When k = n, this is an equality which is known as the principle of inclusion-exclusion.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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4. Prove the Bonferroni inequalities, which state that for any events {A₁, A2,..., An} CF and any integer k € {1,2,..., n}, P(U-1A₁) (≤ Σe=1(-1)²-¹ Σsc{1,2,...,n} P(NjesAj), if k is odd, |S|=l ≥ Σk=1(−1)²-¹ Σsc{1,2,...,n} P(NjesAj), if k is even. |S|=l Do this by expanding 1 {U-₁4;} = 1 − II;_1(1 − 1 {A;}), bound- =1 ing the product (as a function) from above and below, and taking expectation. (You may use without proof that if F G as random variables then EF ≤ EG and E1 {A} = P(A). We have shown this for expectation on countable probability spaces, and it remains true in general.) The special case of k = 1 is known as the union bound. When k = n, this is an equality which is known as the principle of inclusion-exclusion.
Transcribed Image Text:4. Prove the Bonferroni inequalities, which state that for any events {A₁, A2,..., An} CF and any integer k € {1,2,..., n}, P(U-1A₁) (≤ Σe=1(-1)²-¹ Σsc{1,2,...,n} P(NjesAj), if k is odd, |S|=l ≥ Σk=1(−1)²-¹ Σsc{1,2,...,n} P(NjesAj), if k is even. |S|=l Do this by expanding 1 {U-₁4;} = 1 − II;_1(1 − 1 {A;}), bound- =1 ing the product (as a function) from above and below, and taking expectation. (You may use without proof that if F G as random variables then EF ≤ EG and E1 {A} = P(A). We have shown this for expectation on countable probability spaces, and it remains true in general.) The special case of k = 1 is known as the union bound. When k = n, this is an equality which is known as the principle of inclusion-exclusion.
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