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?. In the production of a certain type of copper, two
Chapter 5, Problem 152SE(choose chapter or problem)
In the production of a certain type of copper, two types of copper powder (types \(A \text { and } B\) ) are mixed together and sintered (heated) for a certain length of time. For a fixed volume of sintered copper, the producer measures the proportion \(Y_{1}\) of the volume due to solid copper (some pores will have to be filled with air) and the proportion \(Y_{2}\) of the solid mass due to type A crystals. Assume that appropriate probability densities for \(Y_{1} \text { and } Y_{2}\) are
\(\begin{array}{l}
f_{1}\left(y_{1}\right)=\left\{\begin{array}{ll}
6 y_{1}\left(1-y_{1}\right), & 0 \leq y_{1} \leq 1 \\
0, & \text { elsewhere }
\end{array}\right. \\
f_{2}\left(y_{2}\right)=\left\{\begin{array}{ll}
3 y_{2}^{2}, & 0 \leq y_{2} \leq 1 \\
0, & \text { elsewhere }
\end{array}\right.
\end{array}\)
The proportion of the sample volume due to type A crystals is then \(Y_{1} Y_{2}\). Assuming that \(Y_{1}\) and \(Y_{2}\) are independent, find \(P\left(Y_{1} Y_{2} \leq .5\right)\).
Equation Transcription:
Text Transcription:
A and B
Y_1
Y_2
Y_1 and Y_2
f_1(y_1)={ 6y_1(1-y_1) 0\leq y_1 \leq 1 0, elsewhere
f_2(y_2)={ 3y^2_2 0\leq y_2\leq 1 0, elsewhere
Y_1Y_2
Y_1
Y_2
P(Y_1Y_2\leq 5
Questions & Answers
QUESTION:
In the production of a certain type of copper, two types of copper powder (types \(A \text { and } B\) ) are mixed together and sintered (heated) for a certain length of time. For a fixed volume of sintered copper, the producer measures the proportion \(Y_{1}\) of the volume due to solid copper (some pores will have to be filled with air) and the proportion \(Y_{2}\) of the solid mass due to type A crystals. Assume that appropriate probability densities for \(Y_{1} \text { and } Y_{2}\) are
\(\begin{array}{l}
f_{1}\left(y_{1}\right)=\left\{\begin{array}{ll}
6 y_{1}\left(1-y_{1}\right), & 0 \leq y_{1} \leq 1 \\
0, & \text { elsewhere }
\end{array}\right. \\
f_{2}\left(y_{2}\right)=\left\{\begin{array}{ll}
3 y_{2}^{2}, & 0 \leq y_{2} \leq 1 \\
0, & \text { elsewhere }
\end{array}\right.
\end{array}\)
The proportion of the sample volume due to type A crystals is then \(Y_{1} Y_{2}\). Assuming that \(Y_{1}\) and \(Y_{2}\) are independent, find \(P\left(Y_{1} Y_{2} \leq .5\right)\).
Equation Transcription:
Text Transcription:
A and B
Y_1
Y_2
Y_1 and Y_2
f_1(y_1)={ 6y_1(1-y_1) 0\leq y_1 \leq 1 0, elsewhere
f_2(y_2)={ 3y^2_2 0\leq y_2\leq 1 0, elsewhere
Y_1Y_2
Y_1
Y_2
P(Y_1Y_2\leq 5
ANSWER:
Solution 152SE
Step1 of 2:
Let us consider the random variables denotes the volume due to solid copper and the solid mass due to type A crystals. Assume that appropriate probability densities for Y1 and Y2 are:
And,
We need to find