A device contains two circuits. The second circuit is a backup for the first,so the second is used only when the first has failed. The device fails when and only when the second circuit fails. Let X and Y be the times at which the first and second circuits fail, respectively. X and Y have joint probability density function ( 126e e Oy if 0
A device contains two circuits. The second circuit is a backup for the first,so the second is used only when the first has failed. The device fails when and only when the second circuit fails. Let X and Y be the times at which the first and second circuits fail, respectively. X and Y have joint probability density function ( 126e e Oy if 0
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![**Problem Description: Circuit Failure and Expected Time**
A device contains two circuits. The second circuit is a backup for the first, so the second is used only when the first has failed. The device fails when and only when the second circuit fails.
Let \( X \) and \( Y \) be the times at which the first and second circuits fail, respectively. \( X \) and \( Y \) have the joint probability density function:
\[
f_{XY}(x, y) =
\begin{cases}
126e^{-5x}e^{-9y} & \text{if } 0 < x < y < \infty \\
0 & \text{otherwise}
\end{cases}
\]
**Task:**
Find the expected time at which the device fails.
**Input Section:**
A field is provided for users to enter their solution followed by a "Preview" button.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F132c2b50-c806-45f9-8baf-5357fa898a21%2Fab8dd52f-eb22-4b6a-b46a-3b35fa46919c%2F6l3pgoa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description: Circuit Failure and Expected Time**
A device contains two circuits. The second circuit is a backup for the first, so the second is used only when the first has failed. The device fails when and only when the second circuit fails.
Let \( X \) and \( Y \) be the times at which the first and second circuits fail, respectively. \( X \) and \( Y \) have the joint probability density function:
\[
f_{XY}(x, y) =
\begin{cases}
126e^{-5x}e^{-9y} & \text{if } 0 < x < y < \infty \\
0 & \text{otherwise}
\end{cases}
\]
**Task:**
Find the expected time at which the device fails.
**Input Section:**
A field is provided for users to enter their solution followed by a "Preview" button.
![A company is reviewing tornado damage claims under a farm insurance policy. Let \( X \) be the portion of a claim representing damage to the house, and let \( Y \) be the portion of the same claim representing damage to the rest of the property. The joint density function of \( X \) and \( Y \) is
\[
f_{XY}(x, y) =
\begin{cases}
6(1-x-y) & \text{if } 0 < x \text{ and } 0 < y \text{ and } x+y < 1 \\
0 & \text{otherwise}
\end{cases}
\]
Find the probability that the portion of a claim representing damage to the house is less than 0.32.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F132c2b50-c806-45f9-8baf-5357fa898a21%2Fab8dd52f-eb22-4b6a-b46a-3b35fa46919c%2Frshz54m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A company is reviewing tornado damage claims under a farm insurance policy. Let \( X \) be the portion of a claim representing damage to the house, and let \( Y \) be the portion of the same claim representing damage to the rest of the property. The joint density function of \( X \) and \( Y \) is
\[
f_{XY}(x, y) =
\begin{cases}
6(1-x-y) & \text{if } 0 < x \text{ and } 0 < y \text{ and } x+y < 1 \\
0 & \text{otherwise}
\end{cases}
\]
Find the probability that the portion of a claim representing damage to the house is less than 0.32.
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