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Let $X_1$ and $X_2$ two independent random variables with CDF $F_{X_1}(x_1)$ and $F_{X_2}(x_2)$.

Can we say or it is true that $$P\{X_1\leq \alpha- X_2\leq X_2\}=P\{X\leq \alpha\leq 2X_2\}$$

$$P\{X_1\leq \alpha\leq 2X_2\}=F_{X_1}(\alpha)\left[1-F_{X_2}(\frac{\alpha}{2})\right]$$.

Thanks

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  • $\begingroup$ What is $X$? You are given $X_1$ and $X_2$ but there is no $X$ here. $\endgroup$ – Kavi Rama Murthy Jun 14 at 23:36
  • $\begingroup$ Hi, yes I am sorry I forget it is $X_1$ $\endgroup$ – Monir Jun 14 at 23:38
  • $\begingroup$ hi I am try to avoiding something like that $F_{x_i}(\alpha-x_i).$ $\endgroup$ – Monir Jun 14 at 23:40
  • $\begingroup$ because in my PDFs and CDFs, for example if I use $\int_{x_1=0}^{\alpha}f_{X_1}(x_1)F_{X_2}(\alpha-x_1)$ $\endgroup$ – Monir Jun 14 at 23:43
  • $\begingroup$ You cannot use densities unless you are told that the given random variables have densities $\endgroup$ – Kavi Rama Murthy Jun 14 at 23:46
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Both are wrong . For the first one $X_1=1,X_2=2$ and $\alpha =2$ gives a counterexample. For the second one you need continuity of $F_{X_2}$ at the point $\frac {\alpha} 2$. In general you should take the left hand limit of $F_{X_2}$ at the point $\frac {\alpha} 2$ on RHS.

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  • $\begingroup$ So it probability zero? $\endgroup$ – Monir Jun 14 at 23:45
  • $\begingroup$ Yes, in my example LHS is $0$ but RHS is $1$. $\endgroup$ – Kavi Rama Murthy Jun 14 at 23:47
  • $\begingroup$ So there cases where is not zero or it alwas zeros ($X_1\leq \alpha-X_2$ and $\alpha-X_2\leq X_2$) so we can not express using CDF,? $\endgroup$ – Monir Jun 14 at 23:49
  • $\begingroup$ $X_1 \leq \alpha -X_2$ means $X_1+X_2 \leq \alpha$ so you have to carry out an an integration over a suiatble subset of $\mathbb R^{2}$ w.r.t. the joint distribution. $\endgroup$ – Kavi Rama Murthy Jun 14 at 23:52
  • $\begingroup$ Can I modifies this question to put all my problem or I add new question, because first I was find diffuct to slove this probability $P\{X_1+X_2\leq \alpha, X_1\leq X_2\}$. $\endgroup$ – Monir Jun 14 at 23:54

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