Two random variables are independent if their joint density function
is simply the product of the univariate density functions: h(x, y) ¼
f (x)g(y). Covariance between two variates provides a measure of their
simultaneous change. Consider two variates, X and Y, that have the
means m
X
and m
Y
, respectively. Their covariance equals
Cov(x, y) s
XY
¼ E[(x m
X
)(y m
Y
)] ¼ E[xy] m
X
m
Y
(3:1:13)
Obviously, covariance reduces to variance if X ¼ Y: s
XX
¼ s
X
2
.
Positive covariance between two variates implies that these variates
tend to change simultaneously in the same direction rather than in
opposite directions. Conversely, negative covariance between two
variates implies that when one variate grows, the second one tends
to fall and vice versa. Another popular measure of simultaneous
change is the correlation coefficient
Corr(x, y) ¼ Cov(x:y)=(s
X
s
Y
)(3:1:14)
The values of the correlation coefficient are within the range [ 1, 1].
In the general case with N variates X
1
, ...,X
N
(where N > 2),
correlations among variates are described with the covariance matrix.
Its elements equal
Cov(x
i
,x
j
) s
ij
¼ E[(x
i
m
i
)(x
j
m
j
)] (3:1:15)
3.2 IMPORTANT DISTRIBUTIONS
There are several important probability distributions used in quan-
titative finance. The uniform distribution has a constant value within
the given interval [a, b] and equals zero outside this interval
P
U
¼
0, x < a and x > b
1=(b a), a x b
(3:2:1)
The uniform distribution has the following mean and higher-order
moments
m
U
¼ 0, s
2
U
¼ (b a)
2
=12, S
U
¼ 0, K
eU
¼6=5(3:2:2)
The case with a ¼ 0 and b ¼ 1 is called the standard uniform distribu-
tion. Many computer languages and software packages have a library
function for calculating the standard uniform distribution.
20 Probability Distributions
The binomial distribution is a discrete distribution of obtaining n
successes out of N trials where the result of each trial is true with
probability p and is false with probability q ¼ 1 p (so-called Ber-
noulli trials)
P
B
(n; N, p) ¼ C
Nn
p
n
q
Nn
¼ C
Nn
p
n
(1 p)
Nn
,C
Nn
¼
N!
n!(N n)!
(3:2:3)
The factor C
Nn
is called the binomial coefficient. Mean and higher-
order moments for the binomial distribution are equal, respectively,
m
B
¼ Np, s
2
B
¼ Np(1 p), S
B
¼ (q p)=s
B
,K
eB
¼ (1 6pq)=s
B
2
(3:2:4)
In the case of large N and large (N n), the binomial distribution
approaches the form
P
B
(n) ¼
1
ffiffiffiffiffi
2p
p
s
B
exp [(x m
B
)
2
=2s
2
B
], N !1,(N n) !1 (3:2:5)
that coincides with the normal (or Gaussian) distribution (see 3.2.9). In
the case with p 1, the binomial distribution approaches the Poisson
distribution.
The Poisson distribution describes the probability of n successes in
N trials assuming that the fraction of successes n is proportional to
the number of trials: n ¼ pN
P
P
(n, N) ¼
N!
n!(N n)!
n
N

n
1
n
N

Nn
(3:2:6)
As the number of trials N becomes very large (N !1), equation
(3.2.6) approaches the limit
P
P
(n) ¼ n
n
e
n
=n! (3:2:7)
Mean, variance, skewness, and excess kurtosis of the Poisson distri-
bution are equal, respectively,
m
P
¼ s
2
P
¼ n,S
P
¼ n
1=2
,K
eP
¼ n
1
(3:2:8)
The normal (Gaussian) distribution has the form
P
N
(x) ¼
1
ffiffiffiffiffi
2p
p
s
exp [(x m)
2
=2s
2
](3:2:9)
Probability Distributions 21
It is often denoted N(m, s). Skewness and excess kurtosis of the
normal distribution equals zero. The transform z ¼ (x m)=s con-
verts the normal distribution into the standard normal distribution
P
SN
(z) ¼
1
ffiffiffiffiffi
2p
p
exp [z
2
=2] (3:2:10)
Note that the probability for the standard normal variate to assume
the value in the interval [0, z] can be used as the definition of the error
function erf(x)
1
ffiffiffiffiffi
2p
p
ð
z
0
exp (x
2
=2)dx ¼ 0:5 erf(z=
ffiffi
2
p
)(3:2:11)
Then the cumulative distribution function for the standard normal
distribution equals
Pr
SN
(z) ¼ 0:5[1 þ erf(z=
ffiffi
2
p
)] (3:2:12)
According to the central limit theorem, the probability density distri-
bution for a sum of N independent random variables with finite
variances and finite means approaches the normal distribution as N
grows to infinity. Due to exponential decay of the normal distribu-
tion, large deviations from its mean rarely appear. The normal distri-
bution plays an extremely important role in all kinds of applications.
The Box-Miller method is often used for modeling the normal distri-
bution with given uniform distribution [4]. Namely, if two numbers
x
1
and x
2
are drawn from the standard uniform distribution, then
y
1
and y
2
are the standard normal variates
y
1
¼ [2lnx
1
)]
1=2
cos (2px
2
), y
2
¼ [2lnx
1
)]
1=2
sin (2px
2
)(3:2:13)
Mean and variance of the multivariate normal distribution with N
variates can be easily calculated via the univariate means m
i
and
covariances s
ij
m
N
¼
X
N
i¼1
m
i
, s
2
N
¼
X
N
i
, j¼1
s
ij
(3:2:14)
The lognormal distribution is a distribution in which the logarithm of a
variate has the normal form
22 Probability Distributions

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