probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

, a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

Y

is said to be mean independent of random variable

X

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

its

conditional mean In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on ...

E(Y \mid X = x)

equals its (unconditional)

mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

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 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 Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics", '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

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