Automobile engines and transmissions are produced on assembly lines, and are inspected for defects after they come off their assembly lines. Those with defects are repaired. Let X represent the number of engines, and Y the number of transmissions that require repairs in a one-hour time interval. The joint probability mass function of X and Y is as follows: a. What is the value of C? b. Find the marginal probability ma
Automobile engines and transmissions are produced on assembly lines, and are inspected for defects after they come off their assembly lines. Those with defects are repaired. Let X represent the number of engines, and Y the number of transmissions that require repairs in a one-hour time interval. The joint probability mass
a. What is the value of C?
b. Find the marginal probability mass function .
c. Find the marginal probability mass function .
d. Find u_x .
e. Find u_y.
f. Find sigma_x.
g. Find sigma_y.
h. Find cov(x,y).
i. Find rho_xy.
j. E[X+Y]
k. VAR[X+Y]
l. Are X and Y independent? Explain.
m. What is the CDF Fx(x)?
n. What is the CDF Fy(y)?
![The equation shown is a probability mass function for a discrete random variable \( P_{X,Y}(x,y) \):
\[
P_{X,Y}(x,y) =
\begin{cases}
C e^{-x/3 - y/4} & \text{for } x = 0, 1, 2; \, y = 0, 1, 2 \\
0 & \text{otherwise}
\end{cases}
\]
### Explanation:
- **\( C e^{-x/3 - y/4} \):** The function is defined for the integers \( x \) and \( y \) in the range from 0 to 2, inclusive. The expression involves an exponential function where \( x \) is divided by 3 and \( y \) is divided by 4.
- **\( C \):** This is a normalization constant ensuring that the total probability sums up to 1 over the specified range of \( x \) and \( y \).
- **Outside the specified range for \( x \) and \( y \) (i.e., otherwise),** the probability \( P_{X,Y}(x,y) \) is defined to be 0, indicating that those values are not part of the distribution.
This function could be used in scenarios such as calculating probabilities in statistical models, simulations, or understanding joint probability distributions between two discrete variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6676f4d-69a5-4c0a-bb69-de83bbd667fc%2Fafc66f56-6962-452d-9ef0-b8086d015de6%2Fg5xouta_processed.jpeg&w=3840&q=75)
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