The joint distribution of amount of pollutant emitted from a...

The joint distribution of amount of pollutant emitted from a smokestack without a cleaning device $\left(Y_{1}\right)$ and a similar smokestack with a cleaning device $\left(Y_{2}\right)$ was given in Exercise 5.10 to be $$f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} 1, & 0 \leq y_{1} \leq 2,0 \leq y_{2} \leq 1,2 y_{2} \leq y_{1} \\ 0, & \text { elsewhere } \end{array}\right.$$ The reduction in amount of pollutant due to the cleaning device is given by $U=Y_{1}-Y_{2}$ a. Find the probability density function for $U$ b. Use the answer in part (a) to find $E(U)$. Compare your results with those of Exercise $5.78(\mathrm{c})$.

The joint distribution of amount of pollutant emitted from a smokestack without a cleaning device
$\left(Y_{1}\right)$ and a similar smokestack with a cleaning device $\left(Y_{2}\right)$ was given in Exercise 5.10 to be
$$f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
1, & 0 \leq y_{1} \leq 2,0 \leq y_{2} \leq 1,2 y_{2} \leq y_{1} \\
0, & \text { elsewhere }
\end{array}\right.$$
The reduction in amount of pollutant due to the cleaning device is given by $U=Y_{1}-Y_{2}$
a. Find the probability density function for $U$
b. Use the answer in part (a) to find $E(U)$. Compare your results with those of Exercise $5.78(\mathrm{c})$.

Key Concepts

Joint Probability Density Function

This concept involves a function that assigns probabilities to pairs of outcomes for two random variables. It provides a complete description of the likelihood of different combinations of values, and it forms the basis from which marginal distributions and other properties can be derived, such as when applying transformations.

Transformation of Random Variables

This process refers to finding the distribution of a new random variable that is defined as a function of other random variables. Techniques include changing variables and integrating out the original variables, especially when the function of interest (like the difference between two variables) is used to create a new distribution.

Integration over Restricted Regions

When working with joint distributions, it is common to have conditions or inequalities that define the valid region of integration. Understanding how to set up and evaluate integrals over these regions is crucial for accurately deriving marginal densities and distributions for transformed variables.

Expected Value Calculation

The expected value is a fundamental measure of the central tendency of a random variable. It involves integrating the product of the variable and its probability density function over the entire space. This concept extends to functions of random variables, allowing for comparisons of outcomes such as the net reduction achieved by a cleaning device.

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