Sheets L4
Jointly Distributed Random Variables
Discrete joint distributions:
The joint distribution of the discrete variables X
and Y are given by the joint probability mass
function:
P(X = x and Y = y) for all x SX and y SY
(in a two dimensional table or a formula):
Notation: P(X = x and Y = y)= P(X = x, Y = y)
Requirements:
1. P(X = x and Y = y) ≥ 0
2.
=1
The marginal or individual prob. functions:
P(X = x) =
and
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P(Y = y) =
The conditional probability function of X,
under the condition Y = y, is defined:
P(X = x | Y = y) =
For fixed y,
P(X = x | Y = y) is a probability function
and
the conditional expectation is:
E(X |Y = y) =
Analogously for P(Y = y |X = x) and E(Y|X= x)
Independence of random variables :
Recall: events A and B are
independent  P(A  B) = P(A) P(B)
Two r.v.`s X and Y will be called independent
when every event concerning X and every event
concerning Y are indpendent.
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General definition :
X and Y are independent 
P(XA and YB) = P(XA) P(YB) ,
for any combination A R and B R
For discrete variables this is equivalent to:
X and Y are independent 
P(X = x and Y = y) = P(X = x) P(Y = y)
for all xR en yR
Consequence: if X and Y are independent than
the joint distribution follows logically on the
marginal distributions.
Functions of 2 discrete r.v.
Theorem:
e.g.
It follows that: E(X+Y) = E(X) + E(Y)
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The distribution of Z = X+Y can be found if the
joint probability function is known:
If X and Y are independent, we can write for
the sum or convolution Z = X+Y
(for the case X, Y є {0, 1, 2,…}) :
Convolutions of common discrete distributions:
1. If If X ~ Poisson (µ1), Y ~ Poisson(µ2) and
X and Y are independent,
then X+Y ~ Poisson (µ1 + µ2)
2. If X ~ B(m, p), Y ~ B(n, p) and X and Y are
independent => X + Y ~ B(m+n, p)
The properties above can be extended to n
discrete independent r.v. X1, …, Xn
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X and Y are dependent if they are not
independent
Measure for dependency: covariance
cov(X, Y) = E{(X-EX)(Y-EY)}
For discrete X and Y:
cov(X,Y) =
We say that X and Y are not correlated if
cov(X, Y) = 0
Properties covariance:
1.
2.
3.
4.
cov(X, Y) = E(XY) – EX×EY
cov(X, Y) = cov(Y, X)
cov(X, X) = var(X)
cov(X + a, Y) = cov(X, Y)
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5. cov(aX, Y) = a cov(X, Y)
6. cov(X, Y + Z) = cov(X, Y) + cov(X, Z)
7. var(X+Y) = var(X)+var(Y)+2cov(X,Y) ,
var(X- Y) = var(X)+var(Y) - 2cov(X,Y)
Property 7 for X1, …, Xn :
Properties for independent X and Y :
1. E(XY) = EX×EY
2. cov(X, Y) = 0
3. var(X±Y) = var(X)+var(Y)
Note 1: In general: E(X+Y) = EX+EY ,
but
E(XY) ≠ EX×EY
Note 2 : independence => no correlation
not vice versa !
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The value of the covariance :
- is positive if large values of X in general
coïncide with large values of Y and v.v.
- depends on the unit of measurement
(e.g. 1 m = 100 cm): cov(aX, Y) = a cov(X, Y)
The correlation coefficient ρ(X,Y) does not
depend on the unit of measurement:
Properties ρ(X,Y):
1. -1 ≤ ρ(X,Y) ≤ 1
2. ρ(X,Y) = +1 if for a > 0 P(Y= aX+b) = 1
ρ(X,Y) = -1 if for a < 0 P(Y= aX+b) = 1
3. ρ(aX+b,cY+d) = ρ(X,Y) if a×c > 0
ρ(aX+b,cY+d) = - ρ(X,Y) if a×c < 0
So “ρ(X,Y) is a standardized measure for
linear dependence of X and Y”
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Continuous joint distributions:
If there is a function f(x, y) so that, for every B
R2, P((X,Y) є B) =
, then
(X,Y) are jointly continuous r.v.
Requirements for joint density function f(x,y):
1.
2.
f(x, y) ≥ 0
Notation: f(x, y) = fX,Y(x, y)
The cumulative joint distribution function F
or FX,Y :
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Properties f and F of jointly continuous r.v. :
1. F(x, y) is a continuous function
2. F(x, y) has similar properties as F(x) : non
decreasing, right continuous and similar
limits in both arguments (x and y)
3.
and
4.
5.
and
6. P(X = x and Y = y) = P(X= x) = P(Y = y)= 0
7. Probability of values of (X, Y) in a rectangle:
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Independence of continuous r.v.
General definition (see also discrete joint distr):
X and Y are independent 
P(XA and YB) = P(XA) P(YB) ,
for any combination A R and B R
For continuous variables this is equivalent to:
X and Y are independent 
for all xR en yR
or
for all xR en yR
The distribution of a function of two (or
more) continuous r.v.
The density function f Z(z) of Z = g(X, Y) can
usually be determined in 3 steps:
1. Express FZ(z) = P( g(X,Y) ≤ z) in the joint
or individual distribution of X and Y.
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2. Determine f Z ( z ) 
d
dz
FZ ( z )
3. Use the known distribution.
We will restrict ourselves to simple two
variable functions: X+Y, XY, max(X,Y) and
min(X,Y). And later on we will extend these
properties to n variables X1, …, Xn
Property exponential distribution:
If X ~ E(λ) and Y ~ E(λ) and X and Y are
independent, then min(X,Y) ~ E(2 λ).
The sum or convolution of two independent
continuous r.v. X and Y:
Note: if X and Y are non negative the
boundaries of the integral can be 0 and z.
Consequence for the normal distribution:
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If X ~ N(µ1, σ12) and Y ~ N(µ2, σ22) and X and
Y are independent, then
X +Y ~ N(µ1+ µ2, σ12 + σ22)
For the expectation we can derive:
In particular:
,which can
be used to compute
Many of the definitions and properties for two
variables can be easily extended to 3 or more
variables. Some examples:
- E(X1 +…+ X n) = E(X1) +…+ E(X n)
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If X1,…, Xn are independent, then
- var(X1 +…+ X n) = var(X1) +…+ var(X n)
In statistics we use (relatively) small samples to
get an idea about the population properties.
We model the observed values x1,…, xn in the
sample as random variables X1,…, Xn .
We call X1,…, Xn a random sample if
1. X1,…, Xn are independent.
2. X1,…, Xn all have the same distribution
This population is called the population distribution, so if E(Xi)= µ and var( Xi) = σ2),
Then
 The sum
:
= nµ and
= nσ2 .
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 The sample mean
:
,
 The weak law of large numbers:
for any c > 0
Random samples and the normal distribution
If X1,…, Xn are independent and Xi ~ N(µ, σ2)
(i = 1,…,n) then:
 The sum
 The sample mean
~ N(nµ, nσ2)
~ N(µ,
)
The Central Limit Theorem
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If X1, X2 ,… are independent and E(Xi )= µ and
var(Xi )= σ2, i = 1, 2,… then:
Ф(z)
So for large n (≥ 25) we have approximately :
~ N(nµ, nσ2) and
~ N(µ,
)
Normal approximation of X ~ B(n,p) for large
n (rule np(1-p) > 5 ) : X ~ N(np, np(1-p))
Use continuity correction: P(X≤ k)=P(X≤ k+½)
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