Frequent Used Formula

Graph Stuff

Set

Set Intersection

$A\cap B$ : intersection of A and B (overlapping part, A and B).

Set Union

$A\cup B$ : the union of A and B. (A or B)

Functions

Domain and Range

Inverse Function

Elementary Function

Limit

What is Limit?

Left Hand and Right Hand Limits

Special Limits (very important)

$\lim_{x\rightarrow 0} \frac{sinx}{x} = 1$

For n > 0,

$lim_{x\rightarrow \infty} x^n = \infty$ and $lim_{x\rightarrow \infty} \frac{1}{x^n} = 0$

Continuity

Differentiation Stuff

Derivative

Note: If $f$ is differentiable at $x_0$ , then $f$ is continuous at $x_0$ .

Differentiation

Basic Rules

Chain Rule

Basically : Differential inside, then do outside.

L’Hospital’s Rule

Basically : If it is a fraction and the possible outcome is 0 or $\infty$ when $x = a$ , differentiate both numerator and denominator part.

Probability and Statistics

Three Conceptual Approaches to Probability

Classical Probability

Based on the observations of some experiments that produce outcomes that have equal probability. E.g. tossing a die, flipping a coin.

$P(A) = \frac{Number \space of \space outcomes \space that \space belong \space to \space event \space A}{Total \space number \space of \space outcomes}$

Relative Frequency Concept of Probability

These probabilities cannot be computed using the classical probability rule because the various outcomes for the corresponding experiments are not equally likely. E.g. The probability that an 80-year-old person will live for at least one more year.

$P(A) \approx \frac{f}{n}$

where n is the times if experiment repeated.

Subjective Probability

Many times we face experiments that neither have equally likely outcomes nor can be repeated to generate data. In such cases, we cannot compute the probabilities of events using the classical probability rule or the relative frequency concept. E.g. The probability that Carol, who is taking statistics this
semester, will get an A in this subject.

Counting Rules

Multiplicative Rule
Permutations Rule
Combinations Rule

Conditional Probability

Let A and B be two events and P(B) > 0. The conditional probability of A given B is defined as

$P(A|B) = \frac{P(A\cap B)}{P(B)}$

Bayes’ Theorem

Suppose $B_1$ , · · · , $B_n$ form a partition of the sample space S (that is, they are mutually exclusive and exhaustive), (and P( $B_i$ ) > 0 for any i), then for any k = 1, · · · n and any event A (such that P(A) > 0)

$P\left(B_{k} | A\right)=\frac{P\left(B_{k} \cap A\right)}{P(A)}=\frac{P\left(B_{k}\right) P\left(A | B_{k}\right)}{P\left(B_{1}\right) P\left(A | B_{1}\right)+\cdots+P\left(B_{n}\right) P\left(A | B_{n}\right)}$