Probability and Statistics for Data Science DS-GA 1002 October 10, 2019
Homework 5
Due Sunday, October 20 by 5pm (Submit via Gradescope)
1. Applying rejection sampling. Consider the target pdf:
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0
The following samples are generated iid from an uniform distribution: 0.8235, 0.4387, 0.6948,
0.3816, 0.3171, 0.7655, 0.9502, 0.7952, 0.0344, 0.1869.
a. Use the samples to sample from the target distribution. In order to do this, separate the
samples into pairs: (0.8235, 0.4387), (0.6948, 0.3816), (0.3171, 0.7655), (0.9502, 0.7952),
(0.0344, 0.1869) and then apply rejection sampling.
b. Assume that the target pdf is only nonzero between 0.5 and 1. Find a way of generating
more samples from the available uniform samples. (Hint: Transform some of the samples so
that their pdf is restricted to [0.5, 1].)
c. Another possible way of increasing the number of accepted samples is reordering them to
yield different pairs so that there are less rejections. What is the problem with this?
2. Short questions. Prove these statements or find a counterexample.
a. For any random variable X, E (X2) ≥ E2(X).
b. The median of a random variable X is invariant to shifts, i.e. the median of the random
variable X + b for any constant b, equals median(X) + b.
c. If X and Y have the same distribution and are independent, then E (XY ) = E2
(X).
d. If X1, X2 and X3 have the same distribution,
3. Presents. For Christmas a teacher of a class of n children asks their parents to leave a present
under the Christmas tree in the classroom. The day after each child picks a present at random.
We are interested in computing the expected number of children that end up getting the present
bought by their own parents.
a. What is the pmf for the indicator random variable Ii corresponding to the event kid i gets
the present bought by their own parents?
b. Are Ii and Ij
independent if i 6= j? Justify your answer.
c. What is the expected number of children that end up getting the present bought by their
own parents?
4. Sign flip. We observe a random variable Y given by the product of a continuous random
variable X and a random variable S which equals −1 or 1 with probability 1/2. X and S are
independent.
a. What is the mean of Y ?
b. What is the variance of Y as a function of the mean of X µX and the variance of X σ2X?
c. If X is always nonnegative, compute the pdf of Y as a function of the pdf of X.
d. If the pdf of X is symmetric around the origin, compute the pdf of Y as a function of the pdf of X.