problem 45

4.3.7 problem 45

Notice that

I (A_{1} \cap A_{2} \dots \cap A_{n}) \geq n \sum i = 1 I (A_{i}) - n + 1

because the left-hand side is either $0$ , or $1$ , so the question reduces to whether the left-hand side is ever $0$ , while the right-hand side is $1$ . Notice that this is not possible, becuase if the left-hand side is $0$ , then $A_{j} = 0$ for some $j$ . Thus, $\sum_{i = 1}^{n} I (A_{i}) < n ⟹ R.H.S. < 1$ .

Then,

I (n ⋂ i = 1 A_{i}) \geq n \sum i = 1 I (A_{i}) - n + 1 ⟹ E (I (n ⋂ i = 1 A_{i})) \geq E (n \sum i = 1 I (A_{i}) - n + 1) ⟹ P (n ⋂ i = 1 A_{i}) \geq n \sum i = 1 P (A_{i}) - n + 1

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