# f(X) and f(Y) are identically distributed

Suppose that $$X$$ is a random variable defined on a probability space ( Ω , $$\mathcal F$$ , P )

and $$P_X$$ is the probability measure on ( $$\mathbb{R}$$, $$\mathcal R$$)

and $$Y$$ is a random variable defined on ( Ω' , $$\mathcal F'$$ , P' )

IF $$X$$and $$Y$$ are identically distributed

If $$f:$$ $$\mathbb{R}$$ $$->$$ $$\mathbb{R}$$ is Borel Measurable, And $$X$$ and $$Y$$ are identically distributed, then if we define $$U = f(X)$$ and $$V= f(Y)$$.

$$U$$ and $$V$$ are also identically distributed

(Tried Solution):

After many hints and many hours all I manage to do is the below :

We know $$P_x$$ $$=$$ $$P_Y$$

and $$P_X(B)$$ , $$B \in \mathcal R$$ $$= P_Y(B)$$ , $$B \in \mathcal R$$

$$P(X^{-1}(B)) = P(Y^{-1}(B))$$

$$P(\omega\in Ω : X(ω) \in B) = P(\omega\in Ω' : Y(ω) \in B)$$

Somehow we have to show that the above is equal to the below:. $$......$$ $$......$$

$$P(\omega\in Ω : f o X(ω) \in B) = P(\omega\in Ω' : foY(ω) \in B)$$

Hence, $$P(\omega\in Ω : U(ω) \in B) = P(\omega\in Ω' :V(ω) \in B)$$

Thus, $$U = f(X)$$ and $$V= f(Y)$$ are identically distributed .

Could you please explain the solution thoroughly, because I really want to understand the solution.

• I think you can find the same exercise in Chapter 3.1 Exercise 9 from the textbook of Kai Lai Chung : A Course in Probability Theory – xiao Oct 17 '19 at 8:42

To show that $$f(X)$$ and $$f(Y)$$ are identically distributed, we need to show that for any Borel set $$B$$, we have $$P(f(X) \in B) = P(f(Y) \in B)$$.

We have : $$P(f(X) \in B) = P(\{w : f(X(w)) \in B\}) = P(\{w : X(w) \in f^{-1}(B)\})$$

where $$f^{-1}(B) = \{t \in \mathbb R : f(t) \in B\}$$. Note that this set is also Borel if $$B$$ is, since $$f$$ is Borel measurable. Therefore, all the above probabilities are well defined.

Finally, note that as $$X$$ and $$Y$$ are identically distributed, $$P(\{w : X(w) \in f^{-1}(B)\}) = P_X(f^{-1}(B))= P_Y(f^{-1}(B)) = P(\{w : Y(w) \in f^{-1}(B)\})$$

and now go back using the logic of the first line to $$P(f(Y) \in B)$$ and conclude.

For every Borel set $$B$$:
$$P_U(B)=P_{f(X)}(B)=P(f(X)\in B)=P(X\in f^{-1}(B))=P_X(f^{-1}(B))=$$$$P_Y(f^{-1}(B))=P(Y\in f^{-1}(B))=P(f(Y)\in B)=P_{f(Y)}(B)=P_V(B)$$
The fifth equality rests on the fact that $$X$$ and $$Y$$ have identical distribution.
In the notation above e.g. $$P(X\in A)$$ actually abbreviates $$P(\{X\in A\})$$ where $$\{X\in A\}$$ is on its turn an abbreviation of $$\{\omega\in\Omega\mid X(\omega)\in A\}$$.