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Find the CDF of $Z={X_1+X_2, X_1

Probability $X_1 + X_2+ X_3 geq 1$How to find the probability $P{X_1 + X_2 leq X_3}$?For exponential random variables $X_i$, how to find $P(t-X_1<X_2mid t-X_1<X_3)$?CDF of $X=min{X_1, X_1cdot X_2, X_1cdot X_2cdot X_3, ldots, X_1cdot X_2 cdots X_N}$CDF of $Y=max{X_1, X_2, X_3},$ given $f(x)=2,$ for $0<x<1/2$?Question about the probability $X_1 < X_2$ given $X_2 < X_3$ for IID geometric random variables X?CDF of $Z=X_1+max{X_2,,X_3}$Find $P(X_1 > max(X_2, X_3)| X_1, X_2, X_3 geq d)$Probability $P{min{X_1,X_2}leq X_3}$?Find the probability that $P(X_1leq alpha cap X_2leq X_1)$ and $P(X_1+X_3 leq alpha cap X_1< X_2)$

0

1

$begingroup$

Let $X_1, X_2$ and $X_3$ be three independent exponential random variables. The PDF and CDF of $X_i$ with parameter $beta_i$ are
$$
f_{X_i}(x_i)=beta_i e^{-beta_i x_i}
$$

$$
F_{X_i}(x_i)=1-e^{-beta_i x_i}
$$

What is the CDF of random variable $Z$ define the sum of $X_1$ and $X_2$ where $X_1$ is less then $X_3$.
$$Z={X_1+X_2, X_1<X_3}$$.

Thanks.

probability probability-distributions random-variables

share|cite|improve this question

edited Mar 14 at 13:17

Monir

asked Mar 14 at 12:51

MonirMonir

539

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are your thoughts? What have you tried?
  $endgroup$
  – Easymode44
  Mar 14 at 13:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I want to find the probability that $X_1+X_2leq z$ where $X_1$ is upper bound or less then another random variable $X_3$.
  $endgroup$
  – Monir
  Mar 14 at 13:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, I can see that. Where exactly are you stuck?
  $endgroup$
  – Easymode44
  Mar 14 at 13:22
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I tride $$int_{x_3=0}^{infty}int_{x_2=0}^{x_3}left(int_{x_2=0}^{z-x_1}f_{X_2}(x_2) dx_2right)f_{X_1}(x_1) dx_1f_{X_3}(x_3) dx_2.$$
  $endgroup$
  – Monir
  Mar 14 at 13:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I stuck in the last integral. And i think we need to take all possible cases
  $endgroup$
  – Monir
  Mar 14 at 13:25
  

  
  
  
  

 | 
show 2 more comments

0

1

$begingroup$

Let $X_1, X_2$ and $X_3$ be three independent exponential random variables. The PDF and CDF of $X_i$ with parameter $beta_i$ are
$$
f_{X_i}(x_i)=beta_i e^{-beta_i x_i}
$$

$$
F_{X_i}(x_i)=1-e^{-beta_i x_i}
$$

What is the CDF of random variable $Z$ define the sum of $X_1$ and $X_2$ where $X_1$ is less then $X_3$.
$$Z={X_1+X_2, X_1<X_3}$$.

Thanks.

probability probability-distributions random-variables

share|cite|improve this question

edited Mar 14 at 13:17

Monir

asked Mar 14 at 12:51

MonirMonir

539

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are your thoughts? What have you tried?
  $endgroup$
  – Easymode44
  Mar 14 at 13:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I want to find the probability that $X_1+X_2leq z$ where $X_1$ is upper bound or less then another random variable $X_3$.
  $endgroup$
  – Monir
  Mar 14 at 13:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, I can see that. Where exactly are you stuck?
  $endgroup$
  – Easymode44
  Mar 14 at 13:22
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I tride $$int_{x_3=0}^{infty}int_{x_2=0}^{x_3}left(int_{x_2=0}^{z-x_1}f_{X_2}(x_2) dx_2right)f_{X_1}(x_1) dx_1f_{X_3}(x_3) dx_2.$$
  $endgroup$
  – Monir
  Mar 14 at 13:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I stuck in the last integral. And i think we need to take all possible cases
  $endgroup$
  – Monir
  Mar 14 at 13:25
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

Let $X_1, X_2$ and $X_3$ be three independent exponential random variables. The PDF and CDF of $X_i$ with parameter $beta_i$ are
$$
f_{X_i}(x_i)=beta_i e^{-beta_i x_i}
$$

$$
F_{X_i}(x_i)=1-e^{-beta_i x_i}
$$

What is the CDF of random variable $Z$ define the sum of $X_1$ and $X_2$ where $X_1$ is less then $X_3$.
$$Z={X_1+X_2, X_1<X_3}$$.

Thanks.

probability probability-distributions random-variables

share|cite|improve this question

edited Mar 14 at 13:17

Monir

asked Mar 14 at 12:51

MonirMonir

539

$endgroup$

share|cite|improve this question

edited Mar 14 at 13:17

Monir

asked Mar 14 at 12:51

MonirMonir

539

share|cite|improve this question

edited Mar 14 at 13:17

Monir

asked Mar 14 at 12:51

MonirMonir

539

asked Mar 14 at 12:51

MonirMonir

539

asked Mar 14 at 12:51

MonirMonir

539