In probability theory, a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

Y

is said to be mean independent of random variable

X

if and only if its conditional mean

E(Y ,  X = x)

equals its (unconditional) mean

E(Y)

for all

x

such that the probability density/mass of

X

x

f_X(x)

, is not zero. Otherwise,

Y

is said to be mean dependent on

X

Stochastic independence Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...

implies mean independence, but the converse is not true.; moreover, mean independence implies uncorrelatedness while the converse is not true. Unlike stochastic independence and uncorrelatedness, mean independence is not symmetric: it is possible for

Y

to be mean-independent of

X

even though

X

is mean-dependent on

Y

. The concept of mean independence is often used in econometrics to have a middle ground between the strong assumption of independent random variables (

X_1 \perp X_2

) and the weak assumption of uncorrelated random variables

(\operatorname(X_1, X_2) = 0).

References

Independence (probability theory) {{Probability-stub

Further reading

References