The document outlines fundamental axioms of probability, including nonnegativity, total mass, and the sum of probabilities for disjoint events. It describes various counting methods for probability calculations, including sampling techniques, conditional probabilities, and rules for independent events. Additionally, it covers concepts such as discrete probability distributions, expected value, variance, and joint distributions of random variables, along with properties and theorems such as the Central Limit Theorem.

1. Axioms
Axiom 1: , -
It states that the probability (mass) is nonnegative
Axiom 2: , -
It states that there is a fixed total amount of probability (mass),
namely 1 unit.
Axiom 3: , - , - , -
It states that the total probability (mass) in two disjoint objects
is the sum of the individual probabilities (masses).
Axiom 4:
[⋃ ] ∑ , -
Corollaries
P[A
c
]=1-P[A] 1=P[S]= P[A
c
]+P[A]
P[A]≤1 P[ ]=0
, - , - , -
, -
If A⊂B, then P[A]≤P[B]
, - , - , -
Computing probabilities by counting methods
 Sampling with Replacement and with Ordering:
Number of distinct ordered k-tuples =
 Sampling without Replacement and with Ordering:
Number of distinct ordered k-tuples = ( ) ( )
Permutations of n Distinct Objects
Number of permutations of n objects = ( ) ( )( ) ≜ (We
refer to n! as n factorial.)
 Sampling without Replacement and without Ordering
( ) ( )
( ) ( )
( )
≜( )
 Sampling with Replacement and without Ordering
( ) ( )
Conditional probability
, -
, -
, -
Addition Rule:
, - , - , - , -
If A & B are mutually exclusive then; , - , - , -
Multiplication Rule:
, - , - , - , - , -
If A & B are independent then; , - , - , -
Probability Laws
Total probability law:
, - , - , - , - OR
, - , - , - , - , - , - , - OR
, - , - , - , - , -
Bayes’ Law
, -
, -
, -
, -
, -
, -
, -
, -
, - , -
∑ , - , -
Probability definition:
A & B are mutually exclusive if , -
A & B are independent If , - , - & , - , - &
, - , - , -
Discrete Probability Distributions
Expected value or mean of a discrete random variable X:
, - ∑ ( ) ∑ ( )
Variance of the random variable X
, - ,( ) - ∑( ) ( )
, -
Standard deviation of the random variable X:
, - , - ⁄
X X counts PX Values of x E[X] VAR[X]
Bernoulli
Equals one if the event A
occurs, and zero otherwise.
P0=1-p, p1=p 0,1 p P(1-p)
Binomial Numbers of successes in
fixed n trials
. / ( ) 0,1,. . .,n np np(1-p)
Geometric Number of trials up through
1st success ( ) ( ) 0, 1, .. 1, 2, …
Uniform outcomes are equally likely 1,2,..,L
Poisson number of events that occur
in fixed time period
0,1,2 α α
Probability Formula Sheet Haris H.

2. Cumulative distribution function: ( ) , - , for
Properties
1. ( ) .
2. ( ) .
3. ( ) .
4. ( ) is a non-decreasing function of x, that is, if
a<b, then ( ) ( )
5. ( ) is continuous from the right, that is for h>0,
( ) ( ) ( ).
6. , - ( ) ( )
7. , - ( ) ( )
8. , - ( )
CDF for continuous/discrete random variable:
( ) ∫ ( ) ∫ ∑ ( ) ( )
Probability density function of X:
( )
( )
∑ ( ) ( )
Properties:
( ) ; , - ∫ ( ) ,
( ) ∫ ( ) ; 1 ∫ ( )
Expected value or mean of a random variable X
, - ∫ ( )
Variance of the random variable X:
, - ,( , -) - , - , -
The nth moment of the random variable X:
, - ∫ ( )
Properties of Expected value and Variance:
, - ; , - , - ; , - , -
, - ; , - ; , - , -
, - , -
X X counts FX E[X] VAR[X]
uniform Outcomes with equal
density
{ { ∫
( )
exponential Time until an event { {
Gaussian
(normal)
Number of components
becomes large
( )
√
( )
0
Where
( ) ( )
√
∫ ; ( ) & ( ) ( ) ; ( ) ( ) ( ) ( )
Bounds for probability
 Markov inequality states that
, -
, -
 Chebyshev inequality states that
, - Where is Variance
Pairs of Random Variables
Joint probability mass function:
( ) ,* + * +- ≜ [{ } * +] ( ) ∑ ∑ ( )
Marginal probability mass functions:
( ) [ ] [ ] [* + * + ]
∑ ( ) And similarly for ( ).
Joint CDF of X and Y: ( ) , -
Properties:
 ( ) ( )
 ( ) ( ) ( )  ( ) ( ) ( ) ( )
 , - ( ) ( ) ( ) ( )
Joint PDF of two continuous Random Variables: ( )
( )

3. Marginal pdf: ( ) ∫ ( ) and ( ) ∫ ( )
Properties:
Let ( ) {
1. If x<0 or y<0, then CDF ( )
2. If (x, y) is inside the unit interval,
( ) ∫ ∫
3. Similarly, If and x>1, ( )
4. If and y>1,
( ) ∫ ∫
5. Finally if x>1 and y>1,
( ) ∫ ∫
Independence of two variables:
if X and Y are independent discrete random variables, then the joint pmf is equal to the product of the marginal pmf’s.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Expected Value of a Function of Two Random Variables:
, - {
∫ ∫ ( ) ( )
∑ ∑ ( ) ( )
Joint moment of X and Y:
, - {
∫ ∫ ( )
∑ ∑ ( )
If E[XY]=0, then we say that X and Y are orthogonal.
Covariance of X and Y: COV(X,Y)=E[XY]-E[X]E[Y]
Pairs of independent random variables have covariance zero.
Correlation coefficient of X and Y:
( )
, where √ ( ) √ ( )
X and Y are said to be uncorrelated if
Conditional Probability:
( )
( )
( )
( )
, -
, -
( ) ( )
Conditional Expectation:
, - ∫ ( )
, - ∑ ( )
Central Limit Theorem
Let Sn be the sum of n iid random variables with finite mean E[X]= and finite variance and let Zn be the zero-
mean, unit-variance random variable defined by
Zn
√
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Haris H.