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# The joint distribution of amount of pollutant emitted from

**Chapter 6, Problem 6E**

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**QUESTION:**

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(c).

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### Questions & Answers

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**QUESTION:**

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(c).

**ANSWER:**

Step 1 of 5

A joint probability density function (joint PDF) is a fundamental concept in probability theory and statistics, particularly in the field of multivariate probability distributions. It describes the likelihood of multiple random variables taking specific values simultaneously. The joint PDF is an extension of the concept of a probability density function, which characterizes the probability distribution of a single random variable.

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