Let $\tilde{x}$ and $\tilde{y}$ be random variables with pdfs $f_x(x)$ and $f_y(y)$ and cdfs $F_x(x)$ and $F_y(y)$. Given that
- $E[\tilde{x}] \geq E[\tilde{y}]$
- $F_y(c) \geq F_x(c)$ for all $c \in \mathbb{R}$
How can I show that
$$E[\tilde{x}1_{\tilde{x} < c}]F_y(c) \geq E[\tilde{y}1_{\tilde{y} < c}]F_x(c) $$ for all $c \in \mathbb{R}$. Here, $1_{\tilde{x} < c}$ is an indicator functions that takes on 1 if $\tilde{x} < c$ and 0 otherwise.
Edit:
Per whuber's suggestion, I can use integration by parts to rewrite the problem as
$$F_y(c)\int^c_{-\infty} F_x(x)dx \leq F_x(c)\int^c_{-\infty} F_y(y)dy$$
Relabeling things a bit, I essentially want to show that for two positive functions $g_x$ and $g_y$, $g_y'(c) \geq g_x'(c)$ for all $c \in \mathbb{R}$ implies that
$$g_y'(c)g_x(c) \leq g_x'(c)g_y(c)$$
