Statistics

Question 1

Probability vs Likelihood

Answer 1

Probability:
What is the chance of something occurring given a sample distribution of data
ex. P(height > 170 | mu = 170, sigma = 3.5)

Likelihood:
What is the best distribution of data given a value
ex. P(mu = 170, sigma = 3.5 | height > 170)

Question 2

What are some examples of non-normal distributions?

Answer 2

Uniform: All values equally likely
Exponential: Number of coin flips before heads
Skewed: Long-tailed normal distribution

Question 3

What is a p-value?

Answer 3

P-value:
The probability that the event you observed occurred by random chance

Statistical significance:
When an event did not occur by random chance alone

Question 4

How many ways can you split 10 players in two teams to play 5-a-side football? (Order of teams does not matter)

Answer 4

= (10 choose 5) / 2
= 252 / 2
= 126

252 if order matters
126 if order does not matter

Question 5

What is Bayes theorem? Show formula.

Answer 5

Formula for computing the conditional probability.

P(A|B) = P(B|A)P(A) / P(B)

Question 6

Steps of significance testing

Answer 6

1. Define Null and Alternative Hypothesis
2. Set threshold (alpha)
3. Calculate p-value
4. If p-value < threshold. Reject the null (statistically significant)

Question 7

Combination vs. Permutation? Show formula’s.

Answer 7

Combination. Choose from set of values, order does not matter.
nCm = n! / (n!(n-m)!)

Permutation. Choose from set of values, order matters.
= n! / (n-m)!

Question 8

What is the central limit theorem?

Answer 8

If you take a sufficiently large sample with replacement and take a sampled statistic, and repeat, the distribution of the sample statistic will be normally distributed.

ML relationship:
- Confidence intervals, Comparing models to one another (t-tests, Annova)

Question 9

What are Z-tests, T-tests and ANOVA tests? When would you use each one?

Answer 9

Tests for statistical significance.

T-test (almost always this):
- Normal distribution where we don’t know the standard deviation
- Random sample (<30 points)

Z-test:
- If you know the standard deviation
- Population (>100 points)
- This assumes the variance is identical to any other sample taken from the same population

ANOVA (analysis of variance)
- More than 2 distributions

Question 10

What are the Harmonic and Geometric means?

Answer 10

Harmonic: Average of rates (speeds).
= n / ((1/x1) + (1/x2) + (1/x3))

Geometric: Average of growth rates (bacteria growth, return on investment).
= nthroot(x1x2x3)

Question 11

What is statistical significance? What is a p-value?

Answer 11

Statistical Significance:
- When some event happens not due to chance

P-value
- The probability that the observed event was random chance

Question 12

What is the difference between PDF and CDF?

Answer 12

PDF - The probability that a random variable is exactly x

CDF - The probability that a random variable is less than or equal to x

Question 13

What is entropy? How does this relate to ML?

Answer 13

Entropy is the amount of uncertainty in a random process. (Lower the better)

Using log-loss/categorical cross entropy/binary-cross-entropy all measure entropy. We can use these to measure how well our model makes predictions.

Question 14

MLE vs MAP

Answer 14

Maximum Likelihood Estimate:
- Find the parameters that give us the best distributions for our predictions

Maximum A Posteriori:
- Optimize params, but we have some prior knowledge
- ie. We know the probability in the real world. Given this probability, what are the optimal parameters