?. In the production of a certain type of copper, two

Chapter 5, Problem 152SE

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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

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


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