Page 71 - 4660

3. Finally, the probability of picking two tickets of each colour is

2
( ) ( ) ( ) 2
2
6! 1 1 1 10
P(two of each colour) = = = .
2!2!2! 3 3 3 81
Thus the expected return to any patron was, in pence,
( )
1 4 10
100 + + 40 × = 10.29.
243 81 81
A good time was had by all but the stallholder!

The multivariate Gaussian distribution. A particularly interesting multivariate distribution is
provided by the generalisation of the Gaussian distribution to multiple random variables X i , i = 1,
2, . . . , n. If the expectation value of X i is E(X i ) = µ i then the general form of the PDF is given by
[ ]
1 ∑ ∑
f(x 1 , x 2 , . . . , x n ) = N exp − a ij (x i − µ i )(x j − µ j ) ,
2
i j
where a ij = a ji and N is a normalization constant that we give below. If we write the column
T
T
vectors x = (x 1 x 2 · · · x n ) and µ = (µ 1 µ 2 · · · µ n ) , and denote the matrix with elements a ij by A
T
1
then f(x) = f(x 1 , x 2 , . . . , x n ) = N exp[− (x − µ) A(x − µ), where A is symmetric.
2
We can find that
2
∂ M(0, 0, . . . , 0)
−1
E(X i X j ) = = µ i µ j + (A ) ij ,
∂t i ∂t j
and thus, using (7.11), we obtain
−1
Cov[X i , X j ] = E[(X i − µ i )(X j − µ j )] = (A ) ij .

Hence A is equal to the inverse of the covariance matrix V of the X i , see (7.12). Thus, with the
correct normalization, f(x) is given by

1 1
T
f(x) = exp[− (x − µ) V −1 (x − µ)]. (7.16)
(2π) n/2 (detV ) 1/2 2
In particular, we note that the Y i are independent Gaussian variables with mean zero and
variance λ i . Thus, given a general set of n Gaussian variables x with means µ and covariance
matrix V, one can always perform the above transformation to obtain a new set of variables y,
which are linear combinations of the old ones and are distributed as independent Gaussians with
zero mean and variances λ i .

1.8. The Law of Large Numbers

Markov and Chebyshev’s inequalities

Theorem 8.1. (Markov’s inequality.) If a random variable X accepts only non-negative
values and has a finite mean, then for any positive number α the following inequality holds
E(X)
P(X ≥ α) ≤ . (8.1)
α
⋆

71