Probability theory

Counting

Selecting $k$ elements from $n$ elements can be done in the following number of ways:

	Without replacement	With replacement
Ordered	$n \cdot (n - 1) \cdot . . . (n - k + 1)$	$n^{k}$
Unordered	$(\binom{n}{k})$	$(\binom{n + k - 1}{k})$

Conditional probability is used when we are in a position to update the sample space based on new information.

P (A | B) = \frac{P (A, B)}{P (B)} = \frac{P (B | A) \cdot P (A)}{P (B)}

Let $A_{1}, A_{2}, . . ., A_{n}$ be mutually exclusive events that cover the entire sample space $A$ . Then for any event $B$ ,

P (A_{i} | B) = \frac{P (B | A_{i}) \cdot P (A_{i})}{\sum_{j = 1} P (B | A_{j}) \cdot P (A_{j})}

TIP

The odds form of Bayes' theorem can be also be written as: posterior odds is equal to likelihood ratio times prior odds.

\frac{P (A_{i} | B)}{P (A_{i}^{c} | B)} = \frac{P (B | A_{i})}{P (B | A_{i}^{c})} \cdot \frac{P (A_{i})}{P (A_{i}^{c})}

The survival probability is the probability that a random variable $X$ is greater than a certain value $x$ , denoted as $P (X > x)$ .

The expected value of a function $g (X)$ over a random variable $X \sim f_{X}$ is defined as:

E [g (X)] = \int_{- \infty}^{\infty} g (x) \cdot f_{X} (x) d x