Page 78 - Elements of Distribution Theory

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64 Conditional Distributions and Expectation

the first n − 1games, but, of course, B n cannot depend on D n ,.... We take B 1 = 1so that
initial fortune of the gambler is given by D 1 . The random variables B 1 , B 2 ,... describe
how the gambler uses the information provided by the game to construct her series of bets
and is called a gambling system. The gambler’s fortune after n games is thus
W n+1 = B 1 D 1 + ··· + B n+1 D n+1 .

Then, using the fact that D 1 , D 2 ,... is a martingale difference sequence, and assuming
that B 2 , B 3 ,... are bounded,

E(W n+1 |D 1 ,..., D n ) = D 1 + B 2 D 2 + ··· + B n D n + E(B n+1 D n+1 |D 1 ,..., D n )
= D 1 + B 2 D 2 + ··· + B n D n + B n+1 E(D n+1 |D 1 ,..., D n )
= D 1 + B 2 D 2 + ··· + B n D n ≡ W n .
It follows that E(W n+1 ) = E(W n ), n = 1, 2,... so that
E(W 1 ) = E(W 2 ) = ··· ;
that is, the expected fortune of the gambler after n games is always equal to the initial
fortune. Thus, a gambling system of the type described here cannot convert a fair game into
one that is advantageous to the gambler.

2.7 Exercises
2.1 Let X and Y denote real-valued random variables such that the distribution of (X, Y)is abso-
lutely continuous with density function p(x, y). Suppose that there exist real-valued nonnegative
functions g and h such that
∞ ∞

g(x) dx < ∞ and h(y) dy < ∞
−∞ −∞
and
p(x, y) = g(x)h(y), −∞ < x < ∞, −∞ < y < ∞.
Does it follow that X and Y are independent?
2.2 Let X and Y denote independent random vectors with ranges X and Y, respectively. Consider
functions f :X → R and g:Y → R. Does it follow that f (X) and g(Y) are independent random
variables?
2.3 Let X 1 , X 2 ,..., X m denote real-valued random variables. Suppose that for each n = 1, 2,..., m,
(X 1 ,..., X n−1 ) and X n are independent. Does it follow that X 1 ,..., X m are independent?
2.4 Let X and Y denote real-valued random variables such that the distribution of (X, Y)is absolutely
continuous with density function
1
p(x, y) = , x > 1, y > 1/x.
3 2
x y
Find the density functions of the marginal distributions of X and Y.
2.5 Let X and Y denote real-valued random variables such that the distribution of (X, Y)is discrete
with frequency function
1 e −1 + 2 −(x+y)
p(x, y) = , x, y = 0, 1,....
2e x!y!
Find the frequency functions of the marginal distributions of X and Y.