Show two random variables have the same distribution?

Question

I need help with this question. Thanks :-)! :

Assume that X is uniform on $[0, 1]$ and that $F$ is the cdf of a continuous random variable Y . Show that $Z = F^{−1}(X)$ has the same distribution as $Y$ .

(Note: $X$ uniform on $[a, b]$ means that for any $x \in [a, b]\,\; Pr(X ≤ x) = \frac {(x − a)} {(b − a)}$

Yes. F inverse, so I guess that would mean the quantile function of the random variable Y. — Alexandre de la Tour, Oct 01 '15 at 11:09

score 3 · Accepted Answer · answered Oct 01 '15 at 11:15

3

To demonstrate that $F^{-1}(X)$ and $Y$ are identically distributed, you need to show that: $$\mathsf P(F^{-1}(X)\leq z) = \mathsf P(Y\leq z)$$

You know that $X\sim\mathcal{U}[0;1]$ so then, $\;\mathsf P(X\leq x) = x\;\mathbf 1_{x\in[0;1]}\;$.

And also that the CDF of $Y$ is $\;F(y)\mathop{:=}\mathsf P(Y\leq y) \;$ and $\;F(y)\in[0;1]\;$.

That should get you started.

answered Oct 01 '15 at 11:15

Graham Kemp

129,094

Thank you, that helped a lot. – Alexandre de la Tour Oct 01 '15 at 11:33

Show two random variables have the same distribution?

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