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Properties of joint distributions

3
Now E(X ) is given by
1 1313
3
3
3
3
3
3
3
E(X ) = 1 · p + (2 + 3 + 4 + 5 )p + 6 · 2p = p = 101,
2 2
and the covariance of X and Y is given by
53 253 3660
Cov[X, Y ] = 101 − × = .
13 13 169
The correlation is defined by Corr[X, Y ] = Cov[X, Y ]/σ X σ Y . The standard deviation of Y may
2
be computed from the definition of the variance. Letting µ Y = E(X ) = 253 gives
13
p
2
2
2
2
2
2
2
2
2
2
2
2
2
σ = (1 − µ y ) + p(2 − µ y ) + p(3 − µ y ) + p(4 − µ y ) + p(5 − µ y ) + 2p(6 − µ Y ) =
Y
2
187356 28824
= p = .
169 169
We deduce that
√ √
3660 169 169
Corr[X, Y ] = ≈ 0.984.
169 28824 480
Thus the random variables X and Y display a strong degree of positive correlation, as we would
expect.
We note that the covariance of X and Y occurs in various expressions. For example, if X and Y
are not independent then
2 2 2 2
Var(X + Y ) = E(X + Y ) − (E(X + Y )) = E(X ) + 2E(XY ) + E(Y )−
2
2
−{(E(X)) + 2E(X)E(Y ) + (E(Y )) } = Var(X) + Var(Y ) + 2(E(XY ) − E(X)E(Y )) =
= Var(X) + Var(Y ) + 2Cov[X, Y ].
More generally, we find (for a, b and c constant)
2
2
Var(aX + bY + c) = a Var(X] + b Var(Y ) + 2abCov[X, Y ].
Note that if X and Y are in fact independent then Cov[X, Y ] = 0.
For several variables X i , i = 1, 2, . . . , n, we can define the symmetric (positive definite)
covariance matrix whose elements are

V ij = Cov[X i , X j ], (7.12)

and the symmetric (positive definite) correlation matrix ρ ij = Corr[X i , X j ].

Example 7.4. A card is drawn at random from a normal 52-card pack and its identity
noted. The card is replaced, the pack shuffled and the process repeated. Random
variables W, X, Y, Z are defined as follows:

• W = 2 if the drawn card is a heart; W = 0 otherwise.

• X = 4 if the drawn card is an ace, king, or queen; X = 2 if the card is a
jack or ten; X = 0 otherwise.

• Y = 1 if the drawn card is red; Y = 0 otherwise.

• Z = 2 if the drawn card is black and an ace, king or queen; Z = 0 otherwise.

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