Continuous and Discrete Random Variables
A discrete random variable has positive probabilities associated with a finite number of outcomes.
A continuous random variable has positive probabilities associated with a range of outcome values—the probability of any single value is zero.
Probability Function
A probability function specifies the probability that a random variable is equal to a specific value; P(X = x) = p(x).
A probability density function (pdf) is the expression for a probability function for a continuous random variable.
A cumulative distribution function (cdf) gives the probability of the random variable being equal to or less than each specific value. It is the area under the probability distribution to the left of a specified value.
Discrete Uniform Distribution
A discrete uniform distribution is one where there are n discrete, equally likely outcomes, so that for each outcome p(x) = 1/n.
For a continuous distribution p(x) = 0 for all X; only ranges of value of X have positive probabilities.
Independent and Mutually Exclusive Events
The probability of an independent event is unaffected by the occurrence of other events, but the probability of a dependent event is changed by the occurrence of another event.
Events A and B are independent if and only if:
P(A | B) = P(A), or equivalently, P(B | A) = P(B)
The probability that at least one of two events will occur is P(A or B) = P(A) + P(B) − P(AB). For mutually exclusive events, P(A or B) = P(A) + P(B), since P(AB) = 0.
Joint Probability
The joint probability of two events, P(AB), is the probability that they will both occur. P(AB) = P(A | B) × P(B). For independent events, P(A | B) = P(A), so that P(AB) = P(A) × P(B).
Unconditional Probability and Conditional Probability
Unconditional probability (marginal probability) is the probability of an event occurring.
Conditional probability, P(A | B), is the probability of an event A occurring given that event B has occurred.
