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Introduction to Probability

Exercise 3.20

Answers

(a)

P (X = k) = (\binom{n}{k}) p^{k} {(1 - p)}^{n - k}

for

0 \leq k \leq 3

.

(b)

To use the complement of the desired deven,

P (X > 0) = 1 - P (X = 0) = 1 - {(1 - p)}^{3} = 1 - (- p^{3} + 3 p^{2} - 3 p + 1) = p^{3} - 3 p^{2} + 3 p .

To prove the same by Inclusion-Exclusion,

P (X > 0) = \sum_{i = 1}^{3} P (I_{X_{i}} = 1) - 3 p^{2} + P (∩_{i = 1}^{3} I_{X_{i}} = 1) = 3 p - 3 p^{2} + p^{3} .

(c)

Since

p^{2}

and

p^{3}

go to

0

asymptotically faster than

p

, when

p

is small,

3 p - 3 p^{2} + p^{3} \approx 3 p

.

fifthist

2021-12-05 00:00

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