# Quantile function question

is there a way how to get quantile function from random variable $$Y$$ defined as: $$Y = \begin{cases} 0 \text{ ... with probability 0.25}\\ f(x) \text{ ... with probability 0.75} \end{cases}$$ where $$f(x)$$ is continuous probability distribution function?

Thank you for your time.

Let $$F_Y^{-1}$$ denote the quantile function of $$Y$$. Then $$\forall u\in(0,1),\quad F_Y^{-1}(u)=\left\{\begin{array}{rcl}0&\textrm{if}&u\le\frac14\\f(x)&\textrm{if}&u>\frac14\end{array}\right.$$
If you want the details: Let $$F_Y$$ denote the cumulative distribution function of $$Y$$. Then $$F_Y(y)=0$$ if $$y<0$$, $$F_Y(y)=\frac14$$ if $$0\le y and $$F_Y(y)=1$$ if $$y\ge f(x)$$. You can then deduce the quantile function from its definition: $$F_Y^{-1}(u)=\inf\{y\in\mathbb R\mid F_Y(y)\ge u\}.$$
• I'm sorry I don't understand. Who is $x$? Who is $y$? Are $y$ and $Y$ the same for you? Do you mean that $Y=0$ with probability 0.25 and $f(X)$ with probability 0.75 where $X$ is another random variable? The more you specify, the more I can understand your question and give the answer you're looking for – Will Mar 11 at 19:41