TDSM 2.1

Let [math]A[/math] is the event that people like butter [math]\\Rightarrow P(A) = 0.8[/math]

Probability

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?

(Solution 2.1)

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]

(Solution 2.3)

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

(Solution 2.5)

Statistics

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

(Solution 2.7)

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

(Solution 2.9)

Correlation Analysis

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:

1. 5,000 dollars more than high school grads?
2. 25% more than high school grads?
3. 15% less than high school grads?

(Solution 2.11)

2-13. Use data or literature found in a Google search to estimate/measure the strength of the correlation between:

1. Hits and walks scored for hitters in baseball.
2. Hits and walks allowed by pitchers in baseball.

(Solution 2.13)

Logarithms

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

(Solution 2.15)

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