Probability Distributions and Summary Stats

Covered in 9/12 lecture

Conditional probability: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$

Bayes rule: $P(A\mid B)=\frac{P(B\mid A)\cdot P(A)}{P(B)}$

Expected value: $E[X]=\sum (x_i\cdot p(x_i))$ where $x_i$ is the values that $X$ takes and $p(x_i)$ is the probability that $X$ takes the value $x_i$.

Distributions

• A distribution is a statistical function that gives the probability of a given outcome from an experiment
• Continuous, but large enough sample size converges to the right thing
• Distributions are like histograms

Types:

• Uniform distribution
  • Just a horizontal line
• Normal distribution (Gaussian)
  • Extremely common, often assume normal distribution unless reason to think otherwise
  • Continuous
• Poisson distribution
  • Sums to 1, scrunched up on one side
  • Given the rate of some event occurring, poisson distribution tells you probability of the number of occurrences over a time period
  • Discrete
• Zero-inflated poisson distribution
  • Poisson distribution with a spike at 0
• Bernoulli distribution
  • When you just have one thing with that has a set probability of happening
  • Example: Flipping a coin once
• Binomial distribution
  • The probability of getting a set of outcomes from a set of Bernoulli Trials
  • Discrete
  • Looks like a normal distribution (if you have enough trials)
• Power law distribution
  • e.g. number of close friends

Central Limit Theorem

Central Limit Theorem

If you have a distribution, such as rolling dice (uniform distribution), and you repeatedly sample that and make a new distribution from the means of those samples, then that new distribution will eventually approach a normal distribution with a sufficiently large sample size.

Useful because if you have two funky-looking distributions, you can sample them a bunch of times, and now you have two nice normal distributions that you can compare.

Criteria for samples:

• Picked at random
• Representative of population
• Big enough to draw conclusions (>=30)
• Include less than 10% of the population, if you’re sampling without replacement

Link to original

Summary Statistics

• Examples: Mean, median, mode
• Cannot rely solely on summary statistics, need to first understand your data holistically

Measures of central tendency:

• The Pythagorean means
  • Arithmetic mean - Very sensitive to outliers
  • Geometric mean - A measure of central tendency less sensitive to outliers
  • Harmonic mean - Primarily used for rates (we won’t use it)
• Median
• Mode

Measures of variance:

• Variance: $s^2$
• Standard deviation: $s=\sqrt{\frac{\sum (x_i-\bar{x}{)}^2}{n-1}}$

Other descriptors:

• Skew (left- or right-tailed): Whether distribution is shifted to left or right
  • Negative skew: Mean is shifted to right
  • Positive skew: Mean is shifted to left
• Modality: How the modes are distributed (e.g. unimodal, bimodial, etc.)