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

¼6=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

ﬃﬃﬃﬃﬃﬃ

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

ﬃﬃﬃﬃﬃﬃ

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

ﬃﬃﬃﬃﬃﬃ

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

ﬃﬃﬃﬃﬃﬃ

2p

p

ð

z

0

exp (x

2

=2)dx ¼ 0:5 erf(z=

ﬃﬃﬃ

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=

ﬃﬃﬃ

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

Get *Quantitative Finance for Physicists* now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.