This document provides definitions and properties related to probability theory and statistics. It defines key concepts such as probability spaces, random variables, distribution functions, and probability density functions. It also covers conditional probability, independence, random vectors, and other statistical topics. The document presents the concepts concisely using mathematical notation.

1. Bibliography
Agratini, O., Blaga, P., Coman, Gh., Lectures on Wavelets,
Numerical Methods and Statistics, Casa C˘rtii de Stiint˘,
a¸ ¸ ¸a
Cluj-Napoca, 2005.
Blaga, P., Calculul probabilit˘¸ilor ¸i statistic˘ matematic˘.
at s a a
Vol. II. Curs ¸i culegere de probleme, Universitatea
s
Babe¸-Bolyai, Cluj-Napoca, 1994.
s
Blaga, P., Statistic˘. . . prin Matlab, Presa Universitar˘
a a
Clujean˘, Cluj-Napoca, 2002.
a
Blaga, P., R˘dulescu, M., Calculul probabilit˘¸ilor,
a at
Universitatea Babe¸-Bolyai, Cluj-Napoca, 2002.
s
Lisei, H., Probability theory, Casa C˘rtii de Stiint˘,
a¸ ¸ ¸a
Cluj-Napoca, 2004.
Lisei, H., Micula, S., Soos, A., Probability Theory trough
Problems and Applications, Cluj University Press, 2006.
Shiryaev, A. N., Probability (2nd ed.), Springer, New York
1995.

2. Probability space
E (Experiment) −→ Ω (Outcomes - results of the experiment)
Ω is called sample space
Definition
A non-empty subset K ⊂ P (Ω) is a σ-algebra (σ-field) if it
satisfies the following conditions:
1o if A ∈ K, then A ∈ K (contrary event),
2o if Ai ∈ K, i ∈ I , then the union Ai ∈ K,
i∈I
and the pair (Ω, K) is called the space of events (measurable space
of events).

3. Definition
Let (Ω, K) be a measurable space of events.
The set function
P : K −→ R,
is called probability, if it satisfies the conditions:
1o P (A) 0, ∀ A ∈ K,
2o P (Ω) = 1, Ω is the certain event,
3o P is σ–additive, i.e. for any subset of events Ai ∈ K, i ∈ I ,
pairwise disjoint events (mutually exclusive events),
Ai ∩ Aj = ∅ (impossible event), i = j, it is satisfied
P Ai = P (Ai ) ,
i∈I i∈I
and the triple (Ω, K, P) is called probability space.

4. Proposition
(1) P (∅) = 0,
¯
(2) P A = 1 − P (A),
A) = P (B) − P (A ∩ B),
(3) P (B
A) = P (B) − P (A), if A ⊂ B,
(4) P (B
P (B) , if A ⊂ B,
(5) P (A)
(6) P (A ∪ B) = P (A) + P (B) − P (A ∩ B),
n n
(7) P Ai P (Ai ).
i=1 i=1

5. Property (Poincar´)
e
n n n
P (Ai ) − P (Ai ∩ Aj ) +
P Ai =
i=1 i=1 i,j=1
i<j
n n
n−1
P (Ai ∩ Aj ∩ Ak ) − · · · + (−1)
+ P Ai .
i=1
i,j,k=1
i<j<k

6. Conditional Probability
Definition
Let (Ω, K, P) be a probability space and an event B ∈ K,
P (B) > 0, it is called conditional probability of the event A with
respect to the event B the ratio
P (A ∩ B)
PB (A) = P (A | B) = ·
P (B)

7. Proposition
Let (Ω, K, P) be a probability space, then
(1) for B ∈ K, with P (B) > 0, the function set PB : K −→ R,
defined by
P (A ∩ B)
PB (A) = P (A | B) = , ∀A ∈ K,
P (B)
is a probability, and the triple (Ω, K, PB ) is a probability
space,
(2) ∀A, B ∈ K, satisfying the condition P (A) P (B) > 0, we have
P (A ∩ B) = P (A) P (B | A) = P (B) P (A | B) ,
(3) ∀A ∈ K, satisfying the condition 0 < P (A) < 1, we have
¯ ¯
P (B) = P (A) P (B | A) + P A P B | A , ∀B ∈ K.

8. Property (Multiplication Formula for Probabilities)
Let us consider the events Ai ∈ K, i = 1, n, satisfying
P (∩n Ai ) > 0, then
i=1
n n−1
= P (A1 ) P (A2 | A1 ) P (A3 | A1 ∩ A2 ) . . . P An
P Ai Ai .
i=1 i=1

9. Property (Formula for Total Probability)
Let A ∈ K be an event and a complete set of disjoint events,
(Ai )i∈I , Ai ∈ K, then
P (Ai ) P (A | Ai ) .
P (A) =
i∈I
Property (Bayes’s Formula)
Let us consider (Ai )i∈I , Ai ∈ K, a complete set of disjoint events,
and the event A ∈ K, with P (A) > 0, then
P (Ai ) P (A | Ai )
P (Ai | A) = , for all i ∈ I .
P (Ai ) P (A | Ai )
i∈I

10. Independence
Definition
The events A, B ∈ K are called independent, if
P (A ∩ B) = P (A) P (B) .
Property
Let (Ω, K, P) be a probability space and the events A, B ∈ K,
(1) if P (A) = 0 or P (A) = 1, then the events A and B are
independent,
(2) if the events A and B are independent, then
P (A | B) = P (A), and P (B | A) = P (B),
(3) if the events A and B are independent, then pairs of events
¯ ¯ ¯¯
A, B , A, B and A, B are independent.

11. Definition
The events (Ai )i=1,n are called independent, if for all
1 i1 < i2 < · · · < ik n, and k = 2, n, the following relations are
satisfied
P (Ai1 ∩ Ai2 ∩ · · · ∩ Aik ) = P (Ai1 ) P (Ai2 ) . . . P (Aik ) .
Definition
Let (Ω, K, P) be a probability space. The events of the subset
(Ai )i∈I , Ai ∈ K, are called independent, if
P Aj = P (Aj ) ,
j∈J j∈J
for all finite subsets of indices J ⊂ I .

12. Random Variables
Let (Ω, K, P) a probability space.
Definition
A real function X : Ω −→ R, is called random variable, if
X −1 ((−∞, x)) = (X < x) = ω ∈ Ω X (ω) < x ∈ K,
for all x ∈ R.
Remarks
If |X (Ω)| denotes the cardinal of the range of random variable
X , then the random variable is called:
Discrete random variable, if |X (Ω)| ℵ0 , i.e. it is a
countable (denumerable) set;
Simple random variable, if |X (Ω)| < ℵ0 , i.e. it is a finite
set;
Continuous random variable, if |X (Ω)| = ℵ, i.e. it is a
non-numerable (uncountable) set.

13. Random Vector
Definition
A real valued vector function
X = (X1 , . . ., Xn ) : Ω −→ Rn
is a random vector (n–dimensional random vector), if
(X < x) = (X1 < x1 , . . . , Xn < xn )
= ω∈Ω X1 (ω) < x1 , . . . , Xn (ω) < xn ∈ K,
for all x ∈ Rn .

14. Distribution Function
Let us consider the probability space (Ω, K, P) and a random
variable X : Ω → R.
Definition
The real function
F : R −→ R,
defined by
∀x ∈ R,
F (x) = P (X < x) , (1)
is called the (cumulative) distribution function of X .
Remark
Some books consider this definition for distribution function,
others (Matlab) use the definition given by
∀x ∈ R.
F (x) = P (X x) ,

15. Proposition
Let X be a random variable with the corresponding distribution
function F . Then we have:
(1) ∀ x ∈ R, 0 F (x) 1,
(2) ∀ a, b ∈ R, a < b,
X < b) = F (b) − F (a),
P (a
P (a < X < b) = F (b) − F (a) − P (X = a),
b) = F (b) − F (a) − P (X = a) + P (X = b),
P (a < X
b) = F (b) − F (a) + P (X = b),
P (a X
(3) ∀ x1 , x2 ∈ R, x1 < x2 ,
F (x1 ) F (x2 ) (F is undecreasing function),
(4) lim F (x) = F (−∞) = 0,
x→−∞
(5) lim F (x) = F (+∞) = 1,
x→+∞
(6) ∀ x ∈ R,
F (y ) = F (x − 0) = F (x) (F is left-continuous).
limy x

16. Remark
If F is piecewise constant, then the points of discontinuity xi ,
i ∈ I , are the values of a discrete random variable X , and the
corresponding values of jumps are given by the probability
distribution of X ,
i ∈ I.
pi = P (X = xi ) ,

17. Definition
Let X = (X1 , . . ., Xn ) a random vector.
The real function
F : Rn −→ R,
defined by
∀x ∈ Rn .
F (x) = F (x1 , . . . , xn ) = P (X1 < x1 , . . . , Xn < xn ) ,
is called the (cumulative) distribution function of the random
vector X.
Definition
Let X = (X1 , . . ., Xn ) be a random vector.
All distribution functions of the random vectors (Xi1 , . . ., Xik ),
1 i1 < . . . < ik n, k = 1, n − 1, are called marginal
distribution functions of the random vector X.
Property
If F is the distribution function of random vector X, then the
distribution function of random vector (Xi1 , . . ., Xik ) is given by
Fi1 ,...,ik (xi1 , . . ., xik ) = lim F (x1 , . . ., xn ) .
xj →∞
j=i1 ,...,ik

18. Probability Density Function
Definition
A random variable X is called (absolutely) continuous if the
corresponding distribution function F is absolutely continuous, i.e.
there exists a function f : R → R, such that
x
for all x ∈ R.
F (x) = f (t) dt,
−∞
The function f is called (probability) density function.
Proposition
If the random variable X is continuous, having the distribution
function F and density function f , then:
∀ x ∈ R, f (x)
(1) 0,
(2) F (x) = f (x), a.e. (almost everywhere) on R,
b
(3) for a < b, P (a X < b) = f (x) dx,
a
+∞
(4) f (x) dx = 1.
−∞

19. Remarks
Let X be a continous random variable, then P (X = a) = 0, for
each a ∈ R.
It follows, in this case, that
P (a X < b) = P (a < X b) = P (a < X < b)
b
= P (a X b) = f (x) dx.
a
If the continuous random variable X has the distribution
function F and the density function f , then we have successively
F (x + ∆x) − F (x)
f (x) = F (x) = lim
∆x
∆x→0
P (x X < x + ∆x)
·
= lim
∆x
∆x→0
Therefore, for small values of ∆x, we have
X < x + ∆x) ≈ f (x) ∆x.
P (x

20. Definition
A random vector X = (X1 , . . ., Xn ) is called (absolutely)
continuous if the corresponding distribution function F is
absolutely continuous, i.e. there exists a function f : Rn → R,
called (probability) density function, such that
x1 xn
F (x) = F (x1 , . . ., xn ) = ... f (t1 , . . ., tn ) dt1 . . .dtn ,
−∞ −∞
for all x = (x1 , . . ., xn ) ∈ Rn .
Proposition
Let X = (X1 , . . ., Xn ) be a continuous random vector, having the
distribution function F and the density function f , then
f (x) 0, for all x ∈ Rn ,
(1)
∂ n F (x1 , . . ., xn )
(2) = f (x1 , . . ., xn ), a.e. (almost everywhere)
∂x1 . . . ∂xn
on Rn ,
if D ⊂ Rn , we have P (X ∈ D) =
(3) f (x) dx,
D

21. Definition
Let X = (X1 , . . ., Xn ) a continuous random vector. All densities
functions of the random vectors (Xi1 , . . ., Xik ),
1 i1 < . . . < ik n, k = 1, n − 1, are called marginal densities
functions of X.
Property
Let f be the density function of the continous random vector X,
then the density function of random vector (Xi1 , . . ., Xik ) is given by
···
fi1 ,...,ik (xi1 , . . ., xik ) = f (x1 , . . ., xn ) dxj1 . . .dxjn−k ,
Rn−k
where {j1 , . . ., jn−k } = {1, . . ., n} {i1 , . . ., ik }.
Remark
The density function of Xi is given by
··· xi ∈ R.
fi (xi ) = f (x1 , . . ., xn ) dx1 . . .dxi−1 dxi+1 . . .dxn ,
Rn−1

22. Independence
Definition
Let X = (X1 , . . ., Xn ) be a random vector having distribution
function F . The random variables X1 , . . ., Xn are independent if
F (x) = F (x1 , . . ., xn ) = FX1 (x1 ) . . .FXn (xn ) ,
for all x = (x1 , . . ., xn ) ∈ Rn , FXi being the distribution function of
random variable Xi .

23. Remarks
If X = (X1 , . . ., Xn ) is a discrete random vector, then the
random variables X1 , . . ., Xn are independent if and only if
P (X1 = x1 , . . ., Xn = xn ) = P (X1 = x1 ) . . .P (Xn = xn ) ,
for all xi ∈ Xi (Ω), i = 1, n.
If X = (X1 , . . ., Xn ) is a continuous random vector, then the
random variables X1 , . . ., Xn are independent if and only if the
density function of random vector X satisfies the relation
f (x) = f (x1 , . . ., xn ) = fX1 (x1 ) . . .fXn (xn ) ,
for all x = (x1 , . . ., xn ) ∈ Rn , fXi being the density function of
random variable Xi .

24. Proposition
If the continuous random vector (X , Y ) has the density function f ,
then the density functions for the random variables sum, product,
and ratio are respectively:
+∞
f (u, x − u) du, x ∈ R,
fX +Y (x) =
−∞
+∞
x du
x ∈ R,
fXY (x) = f u, ,
|u|
u
−∞
+∞
f (xu, u) |u| du, x ∈ R.
fX /Y (x) =
−∞
If the random variables X and Y are independent, then
+∞
fX (u) fY (x − u) du, x ∈ R,
fX +Y (x) =
−∞
+∞
x du
x ∈ R,
fXY (x) = fX (u) fY ,
|u|
u
−∞
+∞
fX (xu) fY (u) |u| du, x ∈ R.
fX /Y (x) =
−∞

25. Discrete Uniform Distribution
The random variable X has discrete uniform distribution, when the
distribution is
x 1
where N ∈ N;
X , f (x) = f (x; N) = , x = 1, N.
1 N
N x=1,N
The distribution function is

0, if x 1,



k

F (x) = F (x; N) = , if k < x k + 1, k = 1, N − 1,
N



1, if x > N,

26. The distribution of random variable X is given by
f(x)
1/N
• • •
0 1 2 3 4 N−1 N x
f(x)=1/N, pentru x=1,...,N
The graph of distribution function F is
F(x)
1
(N−1)/N
• •
•
•
•
•
2/N
1/N
• • •
0 1 2 3 N−1 N x

27. Binomial Distribution
The random variable X has binomial distribution, we denote
B (n, p), when the distribution is
x n x n−x
X , f (x) = f (x; n, p) = pq , x = 0, n,
x
f (x) x=0,n
and p ∈ (0, 1), q = 1 − p.
The random variable X represents the number of successes in n
independent trials of an experiment.
Binomial distribution was descovered by James Bernoulli, and was
presented in his book Ars Conjectandi (1713).
1
Pascal considered the particular case p = 2 ·
The distribution function is given by

0, if x 0,



k−1

n i n−i
F (x) = F (x; n, p) = p q , if k − 1 < x k, k = 1, n,
i


 i=0


1, if x > n,

28. In Figure, the vertical bars represent the probabilities of binomial
distribution B (5, 0.4) and the corresponding distribution function.
f(x) F(x)
•
•
•
•
•
•
0 1 2 3 4 5 x
0 1 2 3 4 5 x
f(x)=(n)pxqn−x, pentru x=0,...,n
x

29. Poisson Distribution
The random variable X has Poisson distribution, we denote
Po (λ), when distribution is given by
x λx −λ
X , where f (x) = f (x; λ) = e , and λ > 0.
x!
f (x) x=0,1,2,...
The random variable X , having Poisson distribution, gives the
number of occurrences of a fixed event in a time interval, on a
distance, on an area, and so on.
Poisson (1837) proved that this distribution is a limit case of
binomial distribution.
Namely, when p = p (n) and np → λ, for n → ∞, one obtains that
λx λ
n x n−x
f (x; n, p) = pq e.
x x!
One remarks that for high values of n, p has to be small, to hold
on the product np constant (λ). Thus, the probability p of
considered event is small, when n is high.
It is the reason that this distribution to be called the distribution of
rare events.

30. The Figure illustrates this remark.
It was considered binomial distribution B (100, 0.1) and
corresponding Poisson distribution Po (10), because np = 10.
Probabilitatile distributiilor
0.14
Legea binomiala
Legea lui Poisson
0.12
0.1
0.08
0.06
0.04
0.02
4 6 8 10 12 14 16 18
We also remark a known result, which establishes the connection
between the Poisson distribution and exponential distribution!
When Poisson distribution gives the number of occurrences in a
time interval, the exponential distribution gives the length of the
interval between two successsive occurrences of events.

31. Uniform distribution
The random variable aleatoare X has a uniform distribution on the
interval [a, b], we denote by U (a, b), when the density function is

 1 , if x ∈ [a, b],

f (x) = f (x; a, b) = b − a (2)
if x ∈ [a, b].

0, /
The distribution function of X is 
0, if x < a,


x

x − a
, if a x b,
F (x) = F (x; a, b) = f (t; a, b) dt =
b − a
−∞ 


1, if x > b.
The name is in connection with the fact that if one considers
subintervals of the interval [a, b] with the same length , then the
probability that X belongs to such intervals is / (b − a).

32. The Figure contains the graphs of density function and distribution
function.
f(x) F(x)
1
1/(b−a)
a b x a b x
We remark that distribution function is continuous.
Thus, the values x = a and x = b in formula (2) can be attached
to the cases F (x) = 0 and F (x) = 1, respectively.
It follows that density function can be considered non-zero on [a, b]
or (a, b] or [a, b) or (a, b).
In all cases we have uniform distribution U (a, b).

33. Normal Distribution
The random variable X has normal distribution (Gauss-Laplace
distribution) with the parameters µ ∈ R and σ > 0, we denote
N (µ, σ), when the density function is
(x−µ)2
1 −
f (x) = f (x; µ, σ) = √ e 2σ2 , for all x ∈ R.
σ 2π
The corresponding distribution function is given by
x x (t−µ)2
1 −
f (t; µ, σ) dt = √
F (x) = F (x; µ, σ) = e dt,
2σ 2
σ 2π
−∞ −∞
for all x ∈ R.

34. The graphs of density function and distribution function are given
in the Figure.
The curve which represents the graph of density function f is
called the curve of Gauss.
f(x) F(x)
fm
F(µ+σ)
fi
F(µ)=0.5
F(µ−σ)
µ−σ µ µ+σ
µ−σ µ µ+σ x
x
1
For x = µ is obtained the maximum of f , fm = σ√2π .
The inflexion points of f are x = µ ± σ with the corresponding
value f (µ ± σ) = fi = σ√1 ·
2πe

35. The distribution function of standard normal distribution, N (0, 1),
is x
1 2
− t2
Φ (x) = √ e dt, x ∈ R,
2π −∞
and is called Laplace function.
We remark that the function
x
1 2
− t2
˜ (x) = √ e dt, x ∈ R,
Φ
2π 0
is also called Laplace function.
Between the two Laplace functions the following relation
1˜
Φ (x) = + Φ (x) ,
2
holds.
Using the two Laplace functions we have:
x x (t−µ)2
1 −
f (t; µ, σ) dt = √
F (x; µ, σ) = e 2σ2 dt
σ 2π −∞
−∞
x −µ 1 ˜ x −µ
=Φ = +Φ .
σ 2 σ

36. χ2 distribution (Helmert-Pearson)
The random variable X has χ2 distribution or Helmert-Pearson
distribution, and one denotes χ2 (n), when the density function is
1
 n x
x 2 −1 e− 2 , if x > 0,
 n
n

22 Γ
f (x) = f (x; n) = 2

0, if x 0,

where n ∈ N represents the number of degrees of freedom.

37. The Figure contains the graphs of density function of χ2 (n)
distribution, for some values of parameter n.
f(x;n) n=5
n=10
n=20
f(x;5)
f(x;10)
f(x;20)
3 8 18 x
Property
If the random variables X1 , . . . , Xn are independent, each of them
having the same normal distribution with the parameters µ = 0
and σ = 1, then the random variable
n
2
Yn = Xk ,
k=1
is χ2 (n) distrubuted.

38. t distribution (Student)
The random variable X is t-distributed (Student) distributed,
denoted by T (n), when it has the density function
− n+1
n+1
x2
Γ 2
2
x ∈ R,
f (x) = f (x; n) = √ 1+ ,
n n
nπΓ 2
where n ∈ N represents the number of degrees of freedom.
Gosset (1908) descovered this distribution.
The result was not published at the first time, but using the
pseudonym Student, then it was published.
For n = 1, it is obtained Cauchy distribution:
1
x ∈ R.
f (x) = ,
π (1 + x 2 )

39. The Figure contains the graphs of density of T (n), for some values
of n.
f(x) n=1
n=3
n=20
−5 −4 −3 −2 −1 0 1 2 3 4 5 x
Property
If the random varibales X and Y are independent, X is normally
distributed, and Y having χ2 distribution with n degrees of
freedom, then the random variable
X
T=
Y /n
has the Student distribution with n degrees of freedom.

40. F distribution (Fisher–Snedecor)
The random variable X has F (Fisher-Snedecor) distribution,
denoted by F (m, n), when the density function is

 Γ m+n m m m −1 m − m+n
2 2
2
x2 1+ x , x > 0,


Γ mΓ n n n
f (x) = f (x; m, n) = 2 2


0, x 0,
where m, n ∈ N represent the numbers of degrees of freedom.

41. The Figure contains graphs of density function of F (m, n), for
some values of the parameters m and n.
f(x)
m=4, n=2
m=3, n=10
0.71483
m=7, n=10
0.595 x
Property
If the random variables X and Y are independent, with χ2 (m) and
χ2 (n) distributions, then the random variable
X Y
F=
m n
is F (m, n) distributed.

42. Multidimensional normal distribution
Random vector X = (X1 , . . ., Xn ) has n-dimensional
(nondegenerate) normal distribution, one denotes N (µ, V), when
the density function is
f (x) = f (x1 , . . ., xn ) = f (x; µ, V)
1 1
× exp − (x − µ) V−1 (x − µ) ,
= n 1
2
(2π) 2 [det (V)] 2
for all x ∈ Rn .
V is a positive definite matrix of order n, and µ ∈ Rn .
When n = 2, the density function of the random vector (X , Y ),
having the two-dimensional normal distribution can be put in the
form
1
√ ×
f (x, y ) =
2πσ1 σ2 1 − r 2
(x −µ1 )2 (x −µ1 ) (y −µ2 ) (y −µ2 )2
1
× exp − −2r + ,
2 2
2 (1−r 2 ) σ1 σ2
σ1 σ2
for (x, y ) ∈ R2 .

43. The Figure represents the graph of density function of
two-dimensional normal distribution with µ1 = µ2 = 0, σ1 = 1,
σ2 = 2 ¸i r = −0.5.
s
0.07
0.06
0.05
0.04
f(x,y)
0.03
0.02
0.01
0
6
4
3
2 2
0 1
0
−2
−1
−4 −2
−6 −3
y x
Property
If the random vector (X , Y ) is a two-dimensional normally
distributed, N (µ1 , µ2 ; σ1 , σ2 ; r ), then each of the components X
and Y of random vector are normally distributed: N (µ1 , σ1 ) and
N (µ2 , σ2 ) respectively.

44. Conditional distribution
Let (X , Y ) be a two-dimensional random vector, having the
distribution function F .
Definition
The conditional distribution function of the random variable X
with respect to random variable Y , is the function FX |Y : R → R,
given by
FX |Y (x|y ) = P (X < x | Y = y ) , ∀x ∈ R, y ∈ R, fixed.
Remark
We can rewrite
FX |Y (x|y ) = lim P (X < x | y Y < y + h) ,
h 0
and using the definition of conditional probability
P (X < x, y Y < y + h)
FX |Y (x|y ) = lim
P (y Y < y + h)
h0
F (x, y + h) − F (x, y )
·
= lim
h 0 FY (y + h) − FY (y )

45. Let (X , Y ) be a discret random vector, having the distribution
X Y ... yj ...
. .
. .
. .
xi ... pij ...
. .
. .
. .
with (xi , yj ) ∈ R × R, (i, j) ∈ I × J.
Definition
The conditional distribution of random variable X with respect to
random variable Y has the distribution given by
P (X = xi , Y = yj ) pij
pi|j = P (X = xi | Y = yj ) = (i, j) ∈ I ×J,
= ,
P (Y = yj ) pj
where
j ∈ J.
p j = P (Y = yj ) = pij ,
i∈I

46. We remark that the formula
for all i ∈ I ,
pi = P (X = xi ) = p j pi|j ,
j∈J
holds.
It represents the formula for total probability.
We have also the Bayes’s formula:
pi pj|i
pij
(i, j) ∈ I × J.
pi|j = = ,
pij pi pj|i
i∈I i∈I

47. Let (X , Y ) be a two-dimensional continuous random vector, having
the density function f .
Definition
The conditional density function of random variable X with respect
the random variable Y , is the function given by

 f (x, y ) , if f (y ) = 0,
Y

fY (y )
fX |Y (x | y ) =

0, if fY (y ) = 0.

The formula for total probability and the Bayes’s formula hold:
+∞
∀x ∈ R,
fX (x) = fY (y ) fX |Y (x|y ) dy ,
−∞
fX (x) fY |X (y |x)
∀ (x, y ) ∈ R × R.
fX |Y (x|y ) = ,
fX (x) fY |X (y |x) dx
R

48. Example
Let (X , Y ) be a two-dimensional normally distributed random
vector, N (µ1 , µ2 ; σ1 , σ2 ; r ).
The components X and Y are normally distributed: N (µ1 , σ1 )
and N (µ2 , σ2 ) respectively.
The conditional densities of X with respect to Y and Y with
respect to X are:
(x−m1 (y ))2
1 − 22
e 2(1−r )σ1 , ∀ (x, y ) ∈ R × R,
fX |Y (x|y ) =
r 2)
2π (1 −
σ1
(y −m2 (x))2
1 −
2(1−r 2 )σ 2
fY |X (y |x) = ∀ (y , x) ∈ R × R,
e 2,
2π (1 − r 2 )
σ2
where σ1 σ2
m1 (y ) = µ1 + r (y − µ2 ) , m2 (x) = µ2 + r (x − µ1 ) .
σ2 σ1
We remark that the two conditional densities correspond to the
√
normal distribution: N m1 (y ) , σ1 1 − r 2 and
√
N m2 (x) , σ2 1 − r 2 respectively.

49. In the Figure are presented the graphs of the two conditional
densities functions, when a two-dimensional normal distribution
with µ1 = µ2 = 0, σ1 = 1, σ2 = 2, r = −0.5 is considered, for
three values of the two components: X = −2, 0, 2 and
Y = −2, 0, 2.
X = −2
Y = −2
fX|Y(x|y) X=0
Y=0 fY|X(y|x)
X=2
Y=2
x 6y
−3 −2/3 0 2/3 3 −6 −3−1.5 0 1.5 3