Foundations of Data Science

1. How large must δ be if we wish to have P rob ( s < (1 − δ) m ) < ε?
2. If we wish to have P rob ( s > (1 + δ) m ) < ε?
Exercise 12.17 What is the expected number of flips of a coin until a head is reached?
Assume p is probability of a head on an individual flip. What is value if p=1/2?
Exercise 12.18 Given the joint probability
P(A,B) A=0 A=1
B=0 1/16 1/8
B=1 1/4 9/16
1. What is the marginal probability of A? of B?
2. What is the conditional probability of B given A?
Exercise 12.19 Consider independent random variables x 1 , x 2 , and x 3 , each equal to
zero with probability 1 2 . Let S = x 1 + x 2 + x 3 and let F be event that S ∈ {1, 2}. Conditioning
on F , the variables x 1 , x 2 , and x 3 are still each zero with probability 1 2.
Are they
still independent?
Exercise 12.20 Consider rolling two dice A and B. What is the probability that the sum
S will add to nine? What is the probability that the sum will be 9 if the roll of A is 3?
Exercise 12.21 Write the generating function for the number of ways of producing chains
using only pennies, nickels, and dines. In how many ways can you produce 23 cents?
Exercise 12.22 A dice has six faces, each face of the dice having one of the numbers 1
though 6. The result of a role of the dice is the integer on the top face. Consider two roles
of the dice. In how many ways can an integer be the sum of two roles of the dice.
Exercise 12.23 If a(x) is the generating function for the sequence a 0 , a 1 , a 2 , . . ., for what
sequence is a(x)(1-x) the generating function.
Exercise 12.24 How many ways can one draw n a ′ s and b ′ s with an even number of a ′ s.
Exercise 12.25 Find the generating function for the recurrence a i
a 0 = 1.
= 2a i−1 + i where
Exercise 12.26 Find a closed form for the generating function for the infinite sequence
of prefect squares 1, 4, 9, 16, 25, . . .
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