Solution Found!
Measurements are made on the length and width (in cm) of a
Chapter 2, Problem 15E(choose chapter or problem)
Measurements are made on the length and width (in cm) of a rectangular component. Because of measurement error, the measurements are random variables. Let X denote the length measurement and let Y denote the width measurement. Assume that the probability density function of X is
\(f(x)=\left\{\begin{array}{cl}
10 & 9.95<x<10.05 \\
0 & \text { otherwise }
\end{array}\right.
\)
and that the probability density function of Y is
\(g(y)= \begin{cases}5 & 4.9<y<5.1 \\ 0 & \text { otherwise }\end{cases}\)
Assume that the measurements X and Y are independent.
a. Find \(P(X<9.98)\).
b. Find \(P(Y>5.01)\).
c. Find \(P(X<9.98 \text { and } Y>5.01)\).
d. Find \(\mu_X\).
e. Find \(\mu_Y\).
Equation Transcription:
Text Transcription:
f(x)={_0 otherwise ^10 9.95<x<10.05
g(y)={_0 otherwise ^5 4.9<y<5.1
P(X<9.98)
P(Y>5.01)
P(X<9.98 and Y>5.01)
mu_X
mu_Y
Questions & Answers
QUESTION:
Measurements are made on the length and width (in cm) of a rectangular component. Because of measurement error, the measurements are random variables. Let X denote the length measurement and let Y denote the width measurement. Assume that the probability density function of X is
\(f(x)=\left\{\begin{array}{cl}
10 & 9.95<x<10.05 \\
0 & \text { otherwise }
\end{array}\right.
\)
and that the probability density function of Y is
\(g(y)= \begin{cases}5 & 4.9<y<5.1 \\ 0 & \text { otherwise }\end{cases}\)
Assume that the measurements X and Y are independent.
a. Find \(P(X<9.98)\).
b. Find \(P(Y>5.01)\).
c. Find \(P(X<9.98 \text { and } Y>5.01)\).
d. Find \(\mu_X\).
e. Find \(\mu_Y\).
Equation Transcription:
Text Transcription:
f(x)={_0 otherwise ^10 9.95<x<10.05
g(y)={_0 otherwise ^5 4.9<y<5.1
P(X<9.98)
P(Y>5.01)
P(X<9.98 and Y>5.01)
mu_X
mu_Y
ANSWER:
Solution:
Step 1:
Let X denote the length measurement and let Y denote the width measurement. Here the probability density function of X is
And the density function of Y is
We have to find
- P(X<9.98)
- P(Y> 5.01)
- P(X<9.98 and Y>5.01)
- E(X)
- E(Y)
Step 2:
- We have to find the probability P(X<9.98)
P(X<9.98) =