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In a process of sintering (heating) two types of copper
Chapter 6, Problem 14E(choose chapter or problem)
In a process of sintering (heating) two types of copper powder (see Exercise 5.152), the density function for \(Y_{1}\), the volume proportion of solid copper in a sample, was given by
\(f_{1}\left(y_{1}\right)=\left\{\begin{array}{ll} 6 y_{1}\left(1-y_{1}\right), & 0 \leq y_{1} \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.\)
The density function for \(Y_{2}\), the proportion of type A crystals among the solid copper, was given as
\(f_{2}\left(y_{2}\right)=\left\{\begin{array}{ll} 3 y_{2}^{2}, & 0 \leq y_{2} \leq 1 \\
0, & \text { elsewhere } \end{array}\right.\)
The variable \(U=Y_{1} Y_{2}\) gives the proportion of the sample volume due to type A crystals. If
\(Y_{1} \text { and } Y_{2}\) are independent, find the probability density function for .
Equation Transcription:
{
{
Text Transcription:
Y_1
f_1(y_1)={ 6y_1(1-y_1) 0 \leq y_1 \leq 1 0, elsewhere
Y_2
f_2(y_2)= {3y_2^2, 0 \leq y_2 \leq 1 0, elsewhere
U=Y_1 Y_2
Y_1 and Y_2
Questions & Answers
QUESTION:
In a process of sintering (heating) two types of copper powder (see Exercise 5.152), the density function for \(Y_{1}\), the volume proportion of solid copper in a sample, was given by
\(f_{1}\left(y_{1}\right)=\left\{\begin{array}{ll} 6 y_{1}\left(1-y_{1}\right), & 0 \leq y_{1} \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.\)
The density function for \(Y_{2}\), the proportion of type A crystals among the solid copper, was given as
\(f_{2}\left(y_{2}\right)=\left\{\begin{array}{ll} 3 y_{2}^{2}, & 0 \leq y_{2} \leq 1 \\
0, & \text { elsewhere } \end{array}\right.\)
The variable \(U=Y_{1} Y_{2}\) gives the proportion of the sample volume due to type A crystals. If
\(Y_{1} \text { and } Y_{2}\) are independent, find the probability density function for .
Equation Transcription:
{
{
Text Transcription:
Y_1
f_1(y_1)={ 6y_1(1-y_1) 0 \leq y_1 \leq 1 0, elsewhere
Y_2
f_2(y_2)= {3y_2^2, 0 \leq y_2 \leq 1 0, elsewhere
U=Y_1 Y_2
Y_1 and Y_2
ANSWER:
Solution 14E
Step1 of 2:
Let us consider the random variables denotes two types of copper powder, the density function for is given by:
The density function for , the proportion of type A crystal among the solid copper, was given as:
We need to find the probability density function of U.
Step2 of 2:
Let us assume that are independent so,