# 01 Probability Flashcards

random variable

a random variable attaches a value to each possible outcome of a random process

outcomes

outcomes are the mutually exclusive results of the random process and the set of all potential outcomes is called the sample space

probability distribution

The (marginal) probability distribution is the set of all possible outcomes and their associated probabilities

cumulative probability distribution

The cumulative probability distribution is the probability that the random variable is less than or equal to a particular value

joint distribution

The joint distribution is the probability that two (or more) random variables take on certain values simultaneously

conditional distribution

Conditiona distribution is the distribution of a random variable Y conditional on another random variable X taking on a specific value.

P(Y = y | X = x) = P(X = x, Y = y) / P(X = x)

relevant distributions

normal distribution chi-square distribution student t distribution F distribution Bernoulli distribution

expectations

E(X) = sum(xi fx(xi)) Var(X) = E(X^2) - E(X)^2

E(aX)

aE(X)

E(X + Y)

E(X) + E(Y)

Var(aX)

a^2 Var(X) = b^2 σ^2

Var(aX + bY)

a^2 σ(x)^2 + 2 ab σ(xy) + b^2 σ(y)^2

random sampling

selected at random and i.i.d

i.i.d

independently and identically distributed:

- Same marginal distribution
- The value of Y1 provides no information about the value of Y2

law of large numbers

Under general conditions, the sample average will be close to the population mean with very high probability when the sample is large