View source for Mathematical-preliminaries-TDSM

= Mathematical Preliminaries =


'''Probability'''

<br>2-1.
Suppose 80% of people like peanut butter, 89% like jelly, and 78% like both. Given that a randomly sampled person likes peanut butter, what is the probability that she also likes jelly?

[[TDSM 2.1|(Solution 2.1)]]

<br>2-3.
Consider a game where your score is the maximum value from two dice. Compute the probability of each event from <math>\{1, \ldots, 6\}</math>

[[TDSM 2.3|(Solution 2.3)]]

<br>2-5.
If two binary random variables <i>X</i> and <i>Y</i> are independent, is <math>\bar{X}</math> (the complement of <i>X</i>) and <i>Y</i> also independent? Give a proof or a counterexample.

[[TDSM 2.5|(Solution 2.5)]]


'''Statistics'''

<br>2-7.
Construct a probability distribution where none of the mass lies within one <math>\sigma</math> of the mean.

[[TDSM 2.7|(Solution 2.7)]]

<br>2-9.
Show that the arithmetic mean equals the geometric mean when all terms are the same.

[[TDSM 2.9|(Solution 2.9)]]


'''Correlation Analysis'''

<br>2-11. 
What would be the correlation coefficient between the annual salaries of college and high school graduates at a given company, if for each possible job title the college graduates always made:
<ol type="a">
<li>5,000 dollars more than high school grads?</li>
<li>25% more than high school grads?</li>
<li>15% less than high school grads?</li>
</ol>

[[TDSM 2.11|(Solution 2.11)]]

<br>2-13.
Use data or literature found in a Google search to estimate/measure the strength of the correlation between:
<ol type="a">
<li>Hits and walks scored for hitters in baseball.</li>
<li>Hits and walks allowed by pitchers in baseball.</li>
</ol>

[[TDSM 2.13|(Solution 2.13)]]


'''Logarithms'''

<br>2-15.
Show that the logarithm of any number less than 1 is negative.

[[TDSM 2.15|(Solution 2.15)]]

<br>2-17.
Prove that
<math>x \cdot y = b^{(\log_b x + \log_b y)}</math>

[[TDSM 2.17|(Solution 2.17)]]


'''Implementation Projects'''

<br>2-19.
Find some interesting data sets, and compare how similar their means and medians are? What are the distributions where the mean/median differ on the most?

[[TDSM 2.19|(Solution 2.19)]]


'''Interview Questions'''

<br>2-21.
What is the probability of getting exactly <i>k</i> heads on <i>n</i> tosses, where the coin has a probability of <i>p</i> in coming up heads on each toss?  What about <i>k</i> or more heads?

[[TDSM 2.21|(Solution 2.21)]]

<br>2-23.
At halftime of a basketball game you are offered two possible challenges:
<ol type="a">
<li>Take three shots, and make at least two of them.</li>
<li>Take eight shots, and make at least five of them.</li>
</ol>

[[TDSM 2.23|(Solution 2.23)]]

<br>2-25.
Given a stream of <i>n</i> numbers, show how to select one uniformly at random using only constant storage.  What if you don't know <i>n</i> in advance?

[[TDSM 2.25|(Solution 2.25)]]

<br>2-27.
A person randomly types a 8 digit number in a pocket calculator.  What is the probability that the number looks the same even if the calculator is turned upside down.

[[TDSM 2.27|(Solution 2.27)]]

<br>2-29.
What is A/B testing and how does it work?

[[TDSM 2.29|(Solution 2.29)]]

<br>2-31.
We often say that correlation does not imply causation. What does this mean?

[[TDSM 2.31|(Solution 2.31)]]


'''Kaggle Challenges'''

<br>2-33.
Cause-effect pairs: correlation vs causation.
https://www.kaggle.com/c/cause-effect-pairs

[[TDSM 2.33|(Solution 2.33)]]

<br>2-35.
Predict the fate of animals at a pet shelter
https://www.kaggle.com/c/shelter-animal-outcomes

[[TDSM 2.35|(Solution 2.35)]]