- #1

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- 12

## Homework Statement

Discrete random variables ##X,Y,Z## are mutually independent if for all ##x_i, y_j, z_k##,

$$P(X=x_i \wedge Y=y_j \wedge Z=z_k ) = P(X=x_i)P(Y=y_j)P(Z=z_k )$$

I am trying to show (or trying to understand how someone has shown) that ##X,Y## are also independent as a result of ##X,Y,Z## being mutually independent.

## Homework Equations

## The Attempt at a Solution

It starts of with

$$P(X=x_i \wedge Y=y_j ) = \sum_k P(X=x_i \wedge Y=y_j \wedge Z=z_k )$$

before going using the definition of mutual independence for the three variables to complete the proof. This is the step I don't understand. Why is the probability of getting results ##x_i,y_j## equal to the sum (over ##k##) of probabilities of getting results ##x_i, y_j, z_k##?

Many thanks in advance!