CME 106 – Introduction to Probability and Statistics for Engineers https://stanford.edu/~shervine
VIP Cheatsheet: Probability Remark: for any event B in the sample space, we have P (B) =
n
X
P (B|Ai )P (Ai ).
i=1
Amar Albourm
Shervine Amidi r Extended form of Bayes’ rule – Let {Ai , i ∈ [[1,n]]} be a partition of the sample space.
We have:
August,
August 2022
8, 2018 P (B|Ak )P (Ak )
P (Ak |B) = n
X
P (B|Ai )P (Ai )
i=1
Introduction to Probability and Combinatorics
r Sample space – The set of all possible outcomes of an experiment is known as the sample r Independence – Two events A and B are independent if and only if we have:
space of the experiment and is denoted by S. P (A ∩ B) = P (A)P (B)
r Event – Any subset E of the sample space is known as an event. That is, an event is a set
consisting of possible outcomes of the experiment. If the outcome of the experiment is contained
in E, then we say that E has occurred. Random Variables
r Axioms of probability – For each event E, we denote P (E) as the probability of event E r Random variable – A random variable, often noted X, is a function that maps every element
occuring. By noting E1 ,...,En mutually exclusive events, we have the 3 following axioms: in a sample space to a real line.
n
! n
[ X r Cumulative distribution function (CDF) – The cumulative distribution function F ,
(1) 0 6 P (E) 6 1 (2) P (S) = 1 (3) P Ei = P (Ei ) which is monotonically non-decreasing and is such that lim F (x) = 0 and lim F (x) = 1, is
x→−∞ x→+∞
i=1 i=1 defined as:
F (x) = P (X 6 x)
r Permutation – A permutation is an arrangement of r objects from a pool of n objects, in a
given order. The number of such arrangements is given by P (n, r), defined as: Remark: we have P (a < X 6 B) = F (b) − F (a).
n!
P (n, r) = r Probability density function (PDF) – The probability density function f is the probability
(n − r)! that X takes on values between two adjacent realizations of the random variable.
r Relationships involving the PDF and CDF – Here are the important properties to know
r Combination – A combination is an arrangement of r objects from a pool of n objects, where in the discrete (D) and the continuous (C) cases.
the order does not matter. The number of such arrangements is given by C(n, r), defined as:
P (n, r) n!
C(n, r) = = Case CDF F PDF f Properties of PDF
r! r!(n − r)! X X
(D) F (x) = P (X = xi ) f (xj ) = P (X = xj ) 0 6 f (xj ) 6 1 and f (xj ) = 1
Remark: we note that for 0 6 r 6 n, we have P (n,r) > C(n,r).
xi 6x j
ˆ x ˆ +∞
dF
Conditional Probability (C) F (x) = f (y)dy f (x) = f (x) > 0 and f (x)dx = 1
−∞ dx −∞
r Bayes’ rule – For events A and B such that P (B) > 0, we have:
P (B|A)P (A)
P (A|B) = r Variance – The variance of a random variable, often noted Var(X) or σ 2 , is a measure of the
P (B) spread of its distribution function. It is determined as follows:
Remark: we have P (A ∩ B) = P (A)P (B|A) = P (A|B)P (B). Var(X) = E[(X − E[X])2 ] = E[X 2 ] − E[X]2
r Partition – Let {Ai , i ∈ [[1,n]]} be such that for all i, Ai 6= ∅. We say that {Ai } is a partition
if we have: r Standard deviation – The standard deviation of a random variable, often noted σ, is a
n
measure of the spread of its distribution function which is compatible with the units of the
[ actual random variable. It is determined as follows:
∀i 6= j, Ai ∩ Aj = ∅ and Ai = S p
i=1 σ= Var(X)
Stanford University 1 Winter 2018
VIP Cheatsheet: Probability Remark: for any event B in the sample space, we have P (B) =
n
X
P (B|Ai )P (Ai ).
i=1
Amar Albourm
Shervine Amidi r Extended form of Bayes’ rule – Let {Ai , i ∈ [[1,n]]} be a partition of the sample space.
We have:
August,
August 2022
8, 2018 P (B|Ak )P (Ak )
P (Ak |B) = n
X
P (B|Ai )P (Ai )
i=1
Introduction to Probability and Combinatorics
r Sample space – The set of all possible outcomes of an experiment is known as the sample r Independence – Two events A and B are independent if and only if we have:
space of the experiment and is denoted by S. P (A ∩ B) = P (A)P (B)
r Event – Any subset E of the sample space is known as an event. That is, an event is a set
consisting of possible outcomes of the experiment. If the outcome of the experiment is contained
in E, then we say that E has occurred. Random Variables
r Axioms of probability – For each event E, we denote P (E) as the probability of event E r Random variable – A random variable, often noted X, is a function that maps every element
occuring. By noting E1 ,...,En mutually exclusive events, we have the 3 following axioms: in a sample space to a real line.
n
! n
[ X r Cumulative distribution function (CDF) – The cumulative distribution function F ,
(1) 0 6 P (E) 6 1 (2) P (S) = 1 (3) P Ei = P (Ei ) which is monotonically non-decreasing and is such that lim F (x) = 0 and lim F (x) = 1, is
x→−∞ x→+∞
i=1 i=1 defined as:
F (x) = P (X 6 x)
r Permutation – A permutation is an arrangement of r objects from a pool of n objects, in a
given order. The number of such arrangements is given by P (n, r), defined as: Remark: we have P (a < X 6 B) = F (b) − F (a).
n!
P (n, r) = r Probability density function (PDF) – The probability density function f is the probability
(n − r)! that X takes on values between two adjacent realizations of the random variable.
r Relationships involving the PDF and CDF – Here are the important properties to know
r Combination – A combination is an arrangement of r objects from a pool of n objects, where in the discrete (D) and the continuous (C) cases.
the order does not matter. The number of such arrangements is given by C(n, r), defined as:
P (n, r) n!
C(n, r) = = Case CDF F PDF f Properties of PDF
r! r!(n − r)! X X
(D) F (x) = P (X = xi ) f (xj ) = P (X = xj ) 0 6 f (xj ) 6 1 and f (xj ) = 1
Remark: we note that for 0 6 r 6 n, we have P (n,r) > C(n,r).
xi 6x j
ˆ x ˆ +∞
dF
Conditional Probability (C) F (x) = f (y)dy f (x) = f (x) > 0 and f (x)dx = 1
−∞ dx −∞
r Bayes’ rule – For events A and B such that P (B) > 0, we have:
P (B|A)P (A)
P (A|B) = r Variance – The variance of a random variable, often noted Var(X) or σ 2 , is a measure of the
P (B) spread of its distribution function. It is determined as follows:
Remark: we have P (A ∩ B) = P (A)P (B|A) = P (A|B)P (B). Var(X) = E[(X − E[X])2 ] = E[X 2 ] − E[X]2
r Partition – Let {Ai , i ∈ [[1,n]]} be such that for all i, Ai 6= ∅. We say that {Ai } is a partition
if we have: r Standard deviation – The standard deviation of a random variable, often noted σ, is a
n
measure of the spread of its distribution function which is compatible with the units of the
[ actual random variable. It is determined as follows:
∀i 6= j, Ai ∩ Aj = ∅ and Ai = S p
i=1 σ= Var(X)
Stanford University 1 Winter 2018
