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Expected value of a minimum and maximum of a collection of independent random variables

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1

$begingroup$

Suppose I have two independent random variables $X$ and $Y$ with probability density functions $f_X(x)$, where $0 leq x lt a~$, and $f_Y(y)$ with $0 leq y lt b$.

Let $T = minX, Y$ and $W = maxX, Y$. I'm trying to understand how to calculate $E(T)$ and $E(W)$.

So far I've used the formulas that my textbook provides:

beginalign
F_t(T) = Pr(T leq t) &= 1 - Pr(T gt t) \
&= 1 - Pr( minX, Y gt t) \
&= 1 - ( 1 - F_X(t) )(1-F_Y(t))endalign

Likewise

$$F_W(w) = Pr(W leq w) = Pr(maxX, Y leq w) = F_X(w)F_Y(w)$$

Now to calculate the expected values of each variable, $T$ and $W$, my intention was to use the expected value formulas:

beginalign
E(T) &= int 1-F_T(t)dt & &textand & E(W) &= int 1-F_W(w)dw
endalign

What I seem to be stuck on is determining the domain of both $T$ and $W$.

I've been studying for Exam P and the practice problem solutions don't seem to clarify much. Any help would be much appreciated!

probability expected-value order-statistics

share|cite|improve this question

edited Mar 13 at 22:11

Lee David Chung Lin

4,39341242

asked Mar 13 at 19:15

Ausitn NAusitn N

1

New contributor

Ausitn N is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

0

1

$begingroup$

Suppose I have two independent random variables $X$ and $Y$ with probability density functions $f_X(x)$, where $0 leq x lt a~$, and $f_Y(y)$ with $0 leq y lt b$.

Let $T = minX, Y$ and $W = maxX, Y$. I'm trying to understand how to calculate $E(T)$ and $E(W)$.

So far I've used the formulas that my textbook provides:

beginalign
F_t(T) = Pr(T leq t) &= 1 - Pr(T gt t) \
&= 1 - Pr( minX, Y gt t) \
&= 1 - ( 1 - F_X(t) )(1-F_Y(t))endalign

Likewise

$$F_W(w) = Pr(W leq w) = Pr(maxX, Y leq w) = F_X(w)F_Y(w)$$

Now to calculate the expected values of each variable, $T$ and $W$, my intention was to use the expected value formulas:

beginalign
E(T) &= int 1-F_T(t)dt & &textand & E(W) &= int 1-F_W(w)dw
endalign

What I seem to be stuck on is determining the domain of both $T$ and $W$.

I've been studying for Exam P and the practice problem solutions don't seem to clarify much. Any help would be much appreciated!

probability expected-value order-statistics

share|cite|improve this question

edited Mar 13 at 22:11

Lee David Chung Lin

4,39341242

asked Mar 13 at 19:15

Ausitn NAusitn N

1

New contributor

Ausitn N is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

Suppose I have two independent random variables $X$ and $Y$ with probability density functions $f_X(x)$, where $0 leq x lt a~$, and $f_Y(y)$ with $0 leq y lt b$.

Let $T = minX, Y$ and $W = maxX, Y$. I'm trying to understand how to calculate $E(T)$ and $E(W)$.

So far I've used the formulas that my textbook provides:

beginalign
F_t(T) = Pr(T leq t) &= 1 - Pr(T gt t) \
&= 1 - Pr( minX, Y gt t) \
&= 1 - ( 1 - F_X(t) )(1-F_Y(t))endalign

Likewise

$$F_W(w) = Pr(W leq w) = Pr(maxX, Y leq w) = F_X(w)F_Y(w)$$

Now to calculate the expected values of each variable, $T$ and $W$, my intention was to use the expected value formulas:

beginalign
E(T) &= int 1-F_T(t)dt & &textand & E(W) &= int 1-F_W(w)dw
endalign

What I seem to be stuck on is determining the domain of both $T$ and $W$.

I've been studying for Exam P and the practice problem solutions don't seem to clarify much. Any help would be much appreciated!

probability expected-value order-statistics

share|cite|improve this question

edited Mar 13 at 22:11

Lee David Chung Lin

4,39341242

asked Mar 13 at 19:15

Ausitn NAusitn N

1

New contributor

Ausitn N is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited Mar 13 at 22:11

Lee David Chung Lin

4,39341242

asked Mar 13 at 19:15

Ausitn NAusitn N

1

New contributor

Ausitn N is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited Mar 13 at 22:11

Lee David Chung Lin

4,39341242

asked Mar 13 at 19:15

Ausitn NAusitn N

1

New contributor

Ausitn N is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited Mar 13 at 22:11

Lee David Chung Lin

4,39341242

edited Mar 13 at 22:11

Lee David Chung Lin

4,39341242

asked Mar 13 at 19:15

Ausitn NAusitn N

1

New contributor

Ausitn N is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 13 at 19:15

Ausitn NAusitn N

1