Suppose two continuous random variables $X,Y$ satisfy $$ X \leq Y$$ and the cdf $F(y)$ of $Y$ is known so that for a number $c\in [\min Y,\max Y]$ $$P(Y \leq c)=F(c).$$Since $X \leq Y$,can we logically conclude that $X$ is less than c with probability $F(c)$?Or should we coclude that $P(X \leq c) \geq F(c)? $.Thanks for any clarifications
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Only $P(X \leq c) \geq F(c)$ can be concluded. – Sarvesh Ravichandran Iyer Mar 29 '22 at 09:05
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@SarveshRavichandranIyer could you please explain a bit as to why – AgnostMystic Mar 29 '22 at 09:09
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What do you mean with $c\in [\max Y,\min Y]$? – drhab Mar 29 '22 at 09:12
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Take $X=0,Y=1$ and $c=\frac 1 2$. for a counter-example. – Kavi Rama Murthy Mar 29 '22 at 09:15
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@drhab ,thank you for pointing it out ,I made the edit – AgnostMystic Mar 29 '22 at 09:26
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1But how do you know that random variable $Y$ indeed has a minimum and a maximum? Beware that $Y$ is actually a function and its range could be $\mathbb R$. In that case minimum and maximum both lack. – drhab Mar 29 '22 at 09:29
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Thank you again.In my problem Y has a finite range.I should have mentioned that – AgnostMystic Mar 29 '22 at 13:30