Foundations of Data Science

Exercise 12.10 Consider n samples x 1 , x 2 , . . . , x n from a Gaussian distribution of mean
µ and variance σ. For this distribution m = x 1+x 2 +···+x n
is an unbiased estimator of
n
n∑
µ. If µ is known then 1 (x
n i − µ) 2 is an unbiased estimator of σ 2 . Prove that if we
i=1
n∑
approximate µ by m, then 1 (x
n−1 i − m) 2 is an unbiased estimator of σ 2 .
i=1
Exercise 12.11 Given the distribution
1 √
2π3
e − 1 2( x 3) 2 what is the probability that x >1?
Exercise 12.12 e − x2
2 has value 1 at x = 0 and drops off very fast as x increases. Suppose
we wished to approximate e − x2
2 by a function f(x) where
{ 1 |x| ≤ a
f (x) =
0 |x| > a .
What value of a should we use? What is the integral of the error between f(x) and e − x2
2 ?
Exercise 12.13 Given two sets of red and black balls with the number of red and black
balls in each set shown in the table below.
red black
Set 1 40 60
Set 2 50 50
Randomly draw a ball from one of the sets. Suppose that it turns out to be red. What is
the probability that it was drawn from Set 1?
Exercise 12.14 Why cannot one prove an analogous type of theorem that states p (x ≤ a) ≤
E(x)
a ?
Exercise 12.15 Compare the Markov and Chebyshev bounds for the following probability
distributions
{ 1 x = 1
1. p(x) =
0 otherwise
{ 1/2 0 ≤ x ≤ 2
2. p(x) =
0 otherwise
Exercise 12.16 Let s be the sum of n independent random variables x 1 , x 2 , . . . , x n where
for each i
{ 0 Prob p
x i =
1 Prob 1 − p
428