Description
1. Create 1000 samples from a Gaussian distribution with mean -10 and standard deviation 5.
Create another 1000 samples from another independent Gaussian with mean 10 and standard
deviation 5.
(a) Take the sum of 2 these Gaussians by adding the two sets of 1000 points, point by point,
and plot the histogram of the resulting 1000 points. What do you observe?
(b) Estimate the mean and the variance of the sum.
2. Central Limit Theorem. Let Xi be an iid Bernoulli random variable with value {-1,1}.
Look at the random variable Zn =
1
n
PXi
. By taking 1000 draws from Zn, plot its histogram.
Check that for small n (say, 5-10) Zn does not look that much like a Gaussian, but when n
is bigger (already by the time n = 30 or 50) it looks much more like a Gaussian. Check also
for much bigger n: n = 250, to see that at this point, one can really see the bell curve.
3. Estimate the mean and standard deviation from 1 dimensional data: generate 25,000 samples
from a Gaussian distribution with mean 0 and standard deviation 5. Then estimate the mean
and standard deviation of this gaussian using elementary numpy commands, i.e., addition,
multiplication, division (do not use a command that takes data and returns the mean or
standard deviation).
4. Estimate the mean and covariance matrix for multi-dimensional data: generate 10,000 samples
of 2 dimensional data from the Gaussian distribution
Xi
Yi
∼ N −5
5
,
20 .8
.8 30 . (1)
Then, estimate the mean and covariance matrix for this multi-dimensional data using elementary numpy commands, i.e., addition, multiplication, division (do not use a command that
takes data and returns the mean or standard deviation).
5. Download from Canvas/Files the dataset PatientData.csv.
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Each row is a patient and the last column is the condition that the patient has. Do data
exploration using Pandas and other visualization tools to understand what you can about the
dataset. For example:
(a) How many patients and how many features are there?
(b) What is the meaning of the first 4 features? See if you can understand what they mean.
(c) Are there missing values? Replace them with the average of the corresponding feature
column
(d) How could you test which features strongly influence the patient condition and which
do not?
List what you think are the three most important features.
Written Questions
1. Consider two random variables X,Y that are not independent. Their probabilities of are given
by the following table:
X=0 X=1
Y=0 1/4 1/4
Y=1 1/6 1/3
(a) What is the probability that X = 1?
(b) What is the probability that X = 1 conditioned on Y = 1?
(c) What is the variance of the random variable X?
(d) What is the variance of the random variable X conditioned that Y = 1?
(e) What is E[X3 + X2 + 3 Y
7
|Y = 1|]?
2. Consider the vectors v1 = [1, 1, 1] and v2 = [1, 0, 0]. These two vectors define a 2-dimensional
subspace of R
3
. Project the points P1 = [3, 3, 3], P2 = [1, 2, 3], P3 = [0, 0, 1] on this subspace.
Write down the coordinates of the three projected points. (You can use numpy or a calculator
to do arithmetic if you want).
3. Consider a coin such that probability of heads is 2/3. Suppose you toss the coin 100 times.
Estimate the probability of getting 50 or fewer heads. You can do this in a variety of ways.
One way is to use the Central Limit Theorem. Be explicit in your calculations and tell us
what tools you are using in these.
For help: read this introduction to Pandas http://pandas.pydata.org/pandas-docs/stable/
10min.html and this workflow of exploring features (for a different dataset) https://www.kaggle.
com/cast42/exploring-features
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