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.

  • $\begingroup$ 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 $\endgroup$ – 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.

Now ask questions freely on this answer.

| cite | improve this answer | |

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 $\{\omega\in\Omega\mid X(\omega)\in A\}$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.