Chapter 5: Q.5.2 (page 215)

Show that $E [Y] = \int_{0}^{\infty} P \{Y > y\} d y - \int_{0}^{\infty} P \{Y < - y\} d y$ .
Hint: Show that $\int_{0}^{\infty} P \{Y < - y\} d y = - \int_{- \infty}^{0} x f_{y} (x) d x$
$\int_{0}^{\infty} P \{Y > y\} d y = \int_{0}^{\infty} x f_{y} (x) d x$

Short Answer

Expert verified

Therefore,

$E [Y] = \int_{0}^{\infty} P \{Y > y\} d y - \int_{0}^{\infty} P \{Y < - y\} d y$

Hence Proved.

Step by step solution

Given Information.

$E [Y] = \int_{0}^{\infty} P \{Y > y\} d y - \int_{}^{} P \{Y < - y\} d y$

Explanation.

$E [Y] = \int_{- \infty}^{\infty} x . f_{y} (x) d x = \int_{- \infty}^{0} x . f_{y} (x) d x + \int_{0}^{\infty} x . f_{y} (x) d x$

$= I_{1} + I_{2}$

$I_{1} = \int_{- \infty}^{0} x . f_{y} (x) d x = - \int_{0}^{\infty} x . f_{y} (x) d x$

$= - \int_{0}^{\infty} \int_{- x}^{0} d y . f_{y} (x) d x$

Explanation.

$I_{1, - \infty < - x < - y < 0}$

$I_{1} = - \int_{0}^{\infty} \int_{- \infty}^{y} f_{y} (x) d x d y$

$= - \int_{0}^{\infty} P (Y < - y) d y$

$I_{2} = \int_{0}^{\infty} x . f_{y} (x) d x$

$= \int_{0}^{\infty} \int_{0}^{\infty} d y . f_{y} (x) d x, 0 \leq y \leq α \leq \infty$

Explanation.

$I_{2} = \int_{0}^{\infty} \int_{Y}^{\infty} f_{y} (x) d x d y$

$= \int_{0}^{\infty} P (Y > y) d y$

$E (Y) = \int_{0}^{\infty} P \{Y > y\} d y - \int_{0}^{\infty} P \{Y < - y\} d y$

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Show that $E [Y] = \int_{0}^{\infty} P \{Y > y\} d y - \int_{0}^{\infty} P \{Y < - y\} d y$ .
Hint: Show that $\int_{0}^{\infty} P \{Y < - y\} d y = - \int_{- \infty}^{0} x f_{y} (x) d x$
$\int_{0}^{\infty} P \{Y > y\} d y = \int_{0}^{\infty} x f_{y} (x) d x$

Short Answer

Step by step solution

Given Information.

Explanation.

Explanation.

Explanation.

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Show thatE[Y]=∫0∞PY>ydy-∫0∞PY<-ydy.Hint: Show that ∫0∞PY<-ydy=-∫-∞0xfy(x)dx∫0∞PY>ydy=∫0∞xfy(x)dx

Short Answer

Step by step solution

Given Information.

Explanation.

Explanation.

Explanation.

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Most popular questions from this chapter

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Show that $E [Y] = \int_{0}^{\infty} P \{Y > y\} d y - \int_{0}^{\infty} P \{Y < - y\} d y$ .
Hint: Show that $\int_{0}^{\infty} P \{Y < - y\} d y = - \int_{- \infty}^{0} x f_{y} (x) d x$
$\int_{0}^{\infty} P \{Y > y\} d y = \int_{0}^{\infty} x f_{y} (x) d x$