Download ada04

Multivariate distributions
• Suppose we are measuring 2 or more different
properties of a system
Vr
– e.g. rotational and radial velocities of stars in a cluster
– colours and magnitudes of stars in a cluster
– redshifts and peak apparent magnitudes of distant
supernovae
mv
mv
Independent
Dependent
v sin i
B-V
• To what extent does knowing the value of one
random variable X inform us about the other?
z
Joint distribution of X and Y
• Suppose X and Y are two random variables.
• Their joint probability distribution is f(X,Y)
• Normalisation:
f (X,Y ) dXdY  1

 f (X, Y) dX
f (X)   f (X, Y) dY
• Projection on to f(X), f(Y): f (Y ) 
Y
f(Y)
f(X)
X
Independence and correlation
Y
• Independence:
– knowing about X does not inform about Y
– Definition:
f (X, Y)  f (X) f (Y)
X
• Partial correlation:
Y
– knowing about X tells you about Y
– (but maybe not vice versa)
X
• Correlation:
Y
– knowing X determines Y
X
Linear transformations: scaling
• Scaling a random variable X by a constant a:
– Mean:
aX   f (X) aX dX
 a  f (X) X dX  a X
– Variance:
Var(aX)  [aX  aX ]2
 [aX  a X ]2
 a 2 [X  X ]2  a 2 Var(X)
Linear transformations: addition
• Adding together two random variables X and Y:
X  Y   f (X,Y )(X  Y) dXdY
  f (X, Y)X dXdY   f (X,Y)Y dXdY

 f (X, Y) dYX dX    f (X, Y) dXY dY
  f (X) X dX   f (Y) Y dY
 X  Y
• True for any joint PDF!
Why it works...
• Centre of gravity of a joint PDF has a welldefined position independent of choice of
coordinates.
• e.g. could use either (X,Y) or (X+Y,X-Y):
Y
<Y>
<X>
X
Variance and covariance
• The variance of X+Y depends on whether X and
Y are independent or correlated:
Var(X  Y)  [X  Y  X  Y ]2
 [X  Y  X  Y ]2
 [(X  X )  (Y  Y )]2
 (X  X )2  (Y  Y )2  2(X  X )(Y  Y )
 (X  X )  (Y  Y )  2 (X  X )(Y  Y )
2
2
 Var(X)  Var(Y )  2Cov(X,Y )
Linear transformations of many
variables
• Summation of many scaled random variables:
a X
i
i
i
  ai Xi
i


  ai Xi    a i2 2 Xi     ai a j Cov(Xi , X j )
 i
 i
i ji
2
• Note that we can express this in matrix form.
Correlation
Cov(X,Y)
R(X, Y) 
 (X) (Y)
• Correlation coefficient
R is defined:
R = -1
R=0
1

• Correlation matrix for
Cov(Xi , X j ) .
Rij 
 
several random
 (Xi ) (X j ) .
variables:


.
• Hence variance:
R = +1
.
1
.
.
.
.
1
.
. 

. 
. 

1



  ai Xi     ai a j X i  X j Rij
 i
 i j
2