Probability

• If an event has probability p, we can expect first success in 1/p trials

Bayes' theorem

          P(A) * P(B/A)
P(A/B) = ──────────────
              P(B)

—

Derivationʳ:

P(A ∩ B) is the probability of A times probablity of B given that A has already happened.

P(A ∩ B) = P(A) * P(B/A)

It could also be defined as the probability of B times probablity of A given that B has already happened.

P(A ∩ B) = P(B) * P(A/B)

Equating the two,

P(A) * P(B/A) = P(B) * P(A/B)


                 P(A) * P(B/A)
 =>    P(A/B) = ──────────────
                     P(B)

Probability distribution models

Poisson distribution

Same event that happens multiple times over a time interval.

• Discrete
• Parameter: mean/expectation (λ)

Probability of k events (probability density/mass function):

        λᵏ.e⁻ᵏ
P(k) = ───────
         k!

Poisson density function is not continuous. It's defined only for integer values of k.

See: https://brilliant.org/wiki/poisson-distribution/

Gamma distribution

• Parameters
  • Shape parameter k
  • Rate parameter θ

DBT: Mean is past the midpoint in the graph always??

Normal distribution

• aka Gaussian distribution
• Bell shaped curve
• Continuous ??

Categorical distribution

• Value of random variable is from one among a set of predefined categories ʳ

• Each category has an associated probability

• A generalization of Bernoulli distribution ?? ʳ

• Discrete

• Doesn't have anything to do with category theory

Examples:

• Rolling of a 6-faced die.
  • Each category has equal probability: 1/6
• Coin toss
• Selecting marbles from an urn ʳ
  • Suppose urn has 5 red, 3 green and 2 blue marbles
  • Probabilities of categories are like:
    • Red: 5/10
    • Green: 3/10
    • Blue: 2/10

More

• Exponential
  • For between 2 occurrences of a same event??
• Erlang
• Binomial
• Bernoulli
• Uniform distribution

Central limit theorem

Given a collection of points (of any probability distribution. Need not be normal), if we select k number of points repeatedly with replacement (ie, the k points are considered to be 'put back' after each trial), the mean value of the trials will be normally distributed.

See:

• https://www.scribbr.com/statistics/central-limit-theorem/
• https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_probability/BS704_Probability12.html

Books

• Understanding Probability - Henk Tijms
• A first course in probability - Sheldon Ross
• MIT 6.041SC Probabilistic Systems Analysis and Applied Probability course (MIT OpenCoursWare)
  • https://www.youtube.com/playlist?list=PLUl4u3cNGP60A3XMwZ5sep719_nh95qOe
• http://bayesianthink.blogspot.com/2012/12/the-best-books-to-learn-probability.html#.Ur8uWBVx05k
• https://projects.iq.harvard.edu/stat110