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Expected value of $Z=X_1+X_2$ if $X_1

Conditional Expectation as a random variable of independent rendom variablesFind $E(X_1X_2 mid X_2 X_3)$ for i.i.d. symmetric Bernoulli random variables $X_k$Conditional expectation: $E[X_1 X_2mid X_1 + X_2 X_3]$Compute $E[X_1 mid X_2]$; $E[S_nmid X_1]$; $E[S_n mid S_{n-1}]$Find the value of $mathbb{E}(X_1+X_2+ldots+X_N)$ of i.i.d random variables $X_i$s.Explain why $E[X_1|X_1+X_2] = E[X_2|X_1+X_2]$ if $X_1$, $X_2$ are i.i.d.Verifying calculation inside an expected value problemLet $X_1,X_2,X_3$ be iid. U($0,1$) random variables. Then what will be the value of $E(frac{X_1+X_2}{X_1+X_2+X_3}$)?Throw a dice-expected value.Is expected value $E[X_1;X_1leq X_2]= E[X_1;X_1< X_2]$

0

1

$begingroup$

Let $X_1, X_2$ and $X_3$ three independent random variables with PDF $f_{X_i}(x)$.

I would like to compute the average of random vraible saying $Z$. But here i have two events.
The two events are $phi$ and $bar{phi}$

$$
phi={X_3leq X_1},
$$

and

$$bar{phi}={X_1< X_3}.$$

I would like to get the expected value of $Z=X_1$ if the event $phi$ occur

$$
E[X_1 ;
phi],
$$
and the expected value of $Z=X_1+X_2$ if the event $bar{phi}$ occur

$$
E[X_1+X_2;
bar{phi}].
$$

Finally I get
$$
E[Z]=E[X_1;
{phi}]+E[X_1+X_2;
bar{phi}].
$$
Thanks.

conditional-expectation expected-value

share|cite|improve this question

edited Mar 10 at 13:20

Monir

asked Mar 10 at 1:52

MonirMonir

419

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $newcommand{I}{operatorname{mathbb{I}}}newcommand{P}{operatorname{mathbb{P}}}newcommand{E}{operatorname{mathbb{E}}}$What you want is not $color{red}{cap}$, but rather $color{blue}{I}$ (indicator function). That is, $Eleft[Xmid Aright] = dfrac{E[X I(A)]}{P(A)}$. For example, $E[X_1 + X_2 mid X_1 < X_3] = dfrac{Eleft[(X_1 + X_2)I(X_1 < X_3)right]}{P(X_1 < X_3)}$.
  $endgroup$
  – Minus One-Twelfth
  Mar 10 at 2:54
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hi I did not explain every thing, But now I add all my problem.
  $endgroup$
  – Monir
  Mar 10 at 3:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  i am not a statician or mathematician but i want to share my thoughts in a primitive way. we are asked average of x1 given that x1>=x3.. so we are asked average of x1 but we are sure that minimum value it can take is x3. so answer is integration of function from x3 to max value. in second part, because these are random variables, x2 does not care about any relation between x1 and x3. so answer is e(x2) + integral of function from min value to x3... this comment may not fit for standards of this site and it may be wrong, but still i wanted to write it
  $endgroup$
  – xcvbnm
  Mar 10 at 3:24
  

  
  
  
  

add a comment | 

0

1

$begingroup$

Let $X_1, X_2$ and $X_3$ three independent random variables with PDF $f_{X_i}(x)$.

I would like to compute the average of random vraible saying $Z$. But here i have two events.
The two events are $phi$ and $bar{phi}$

$$
phi={X_3leq X_1},
$$

and

$$bar{phi}={X_1< X_3}.$$

I would like to get the expected value of $Z=X_1$ if the event $phi$ occur

$$
E[X_1 ;
phi],
$$
and the expected value of $Z=X_1+X_2$ if the event $bar{phi}$ occur

$$
E[X_1+X_2;
bar{phi}].
$$

Finally I get
$$
E[Z]=E[X_1;
{phi}]+E[X_1+X_2;
bar{phi}].
$$
Thanks.

conditional-expectation expected-value

share|cite|improve this question

edited Mar 10 at 13:20

Monir

asked Mar 10 at 1:52

MonirMonir

419

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $newcommand{I}{operatorname{mathbb{I}}}newcommand{P}{operatorname{mathbb{P}}}newcommand{E}{operatorname{mathbb{E}}}$What you want is not $color{red}{cap}$, but rather $color{blue}{I}$ (indicator function). That is, $Eleft[Xmid Aright] = dfrac{E[X I(A)]}{P(A)}$. For example, $E[X_1 + X_2 mid X_1 < X_3] = dfrac{Eleft[(X_1 + X_2)I(X_1 < X_3)right]}{P(X_1 < X_3)}$.
  $endgroup$
  – Minus One-Twelfth
  Mar 10 at 2:54
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hi I did not explain every thing, But now I add all my problem.
  $endgroup$
  – Monir
  Mar 10 at 3:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  i am not a statician or mathematician but i want to share my thoughts in a primitive way. we are asked average of x1 given that x1>=x3.. so we are asked average of x1 but we are sure that minimum value it can take is x3. so answer is integration of function from x3 to max value. in second part, because these are random variables, x2 does not care about any relation between x1 and x3. so answer is e(x2) + integral of function from min value to x3... this comment may not fit for standards of this site and it may be wrong, but still i wanted to write it
  $endgroup$
  – xcvbnm
  Mar 10 at 3:24
  

  
  
  
  

add a comment | 

$begingroup$

Let $X_1, X_2$ and $X_3$ three independent random variables with PDF $f_{X_i}(x)$.

I would like to compute the average of random vraible saying $Z$. But here i have two events.
The two events are $phi$ and $bar{phi}$

$$
phi={X_3leq X_1},
$$

and

$$bar{phi}={X_1< X_3}.$$

I would like to get the expected value of $Z=X_1$ if the event $phi$ occur

$$
E[X_1 ;
phi],
$$
and the expected value of $Z=X_1+X_2$ if the event $bar{phi}$ occur

$$
E[X_1+X_2;
bar{phi}].
$$

Finally I get
$$
E[Z]=E[X_1;
{phi}]+E[X_1+X_2;
bar{phi}].
$$
Thanks.

conditional-expectation expected-value

share|cite|improve this question

edited Mar 10 at 13:20

Monir

asked Mar 10 at 1:52

MonirMonir

419

$endgroup$

share|cite|improve this question

edited Mar 10 at 13:20

Monir

asked Mar 10 at 1:52

MonirMonir

419

share|cite|improve this question

edited Mar 10 at 13:20

Monir

asked Mar 10 at 1:52

MonirMonir

419

asked Mar 10 at 1:52

MonirMonir

419

asked Mar 10 at 1:52

MonirMonir

419