Probability Theory and Probability Distributions
Core concepts
You must be fluent with the axioms of probability, conditional probability, independence, and Bayes' theorem, and with the standard distributions: binomial, Poisson, geometric, hypergeometric, uniform, normal, exponential, gamma, and their means and variances. Know when a distribution is the right model: counts of rare events (Poisson), fixed number of trials with constant success probability (binomial), sampling without replacement (hypergeometric), waiting times (geometric/exponential). Understand the central limit theorem, the law of large numbers, moment generating functions, and transformations of variables.
Traps
- Confusing P(A|B) with P(B|A); always write the conditioning event explicitly.
- Assuming independence when events are only mutually exclusive (or vice versa).
- Using the binomial when trials are not independent, e.g. sampling without replacement from a small population.
- Forgetting the continuity correction when approximating a discrete distribution by the normal.
- Mixing up the parameterisation of the exponential (rate vs mean) or the variance of the Poisson (it equals the mean).
- Applying the CLT to the original variable rather than to a sum or mean.
Worked example
A call centre receives calls at a mean rate of 3 per minute (Poisson). Find the probability of at most one call in a given minute. P(X=0)=e^{-3}=0.0498; P(X=1)=3e^{-3}=0.1494; total = 0.1991. Rule of thumb: for a Poisson with mean λ, use the normal approximation with continuity correction only when λ is large (roughly λ ≥ 10), setting mean and variance both equal to λ.