In Exercise 5.9, we determined that f(y1, y2)={ 6(1-y2), 0 ≤ y1≤ y2≤ 1 0, else

step 1 In this question we have a pdf f of y1 and y2 is 6 times 1 minus y2 when y1 and y2 are in this r

In Exercise 5.9, we determined that f(y1, y2)={ 6(1-y2),...

In Exercise $5.9,$ we determined that $$f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} 6\left(1-y_{2}\right), & 0 \leq y_{1} \leq y_{2} \leq 1 \\ 0, & \text { elsewere } \end{array}\right.$$ is a valid joint probability density function. In Exercise 5.76 , we derived the fact that $E\left(Y_{1}-3 Y_{2}\right)=-5 / 4 ;$ in Exercise $5.92,$ we proved that $\operatorname{Cov}\left(Y_{1}, Y_{2}\right)=1 / 40 .$ Find $V\left(Y_{1}-3 Y_{2}\right)$

In Exercise $5.9,$ we determined that $$f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
6\left(1-y_{2}\right), & 0 \leq y_{1} \leq y_{2} \leq 1 \\
0, & \text { elsewere }
\end{array}\right.$$
is a valid joint probability density function. In Exercise 5.76 , we derived the fact that $E\left(Y_{1}-3 Y_{2}\right)=-5 / 4 ;$ in Exercise $5.92,$ we proved that $\operatorname{Cov}\left(Y_{1}, Y_{2}\right)=1 / 40 .$ Find $V\left(Y_{1}-3 Y_{2}\right)$

Key Concepts

Variance of Linear Combinations

The variance of a linear combination of random variables is determined by the variances of the individual variables and their covariances. For a linear combination such as aY? + bY?, the variance is given by a²Var(Y?) + b²Var(Y?) + 2abCov(Y?, Y?). This property allows one to compute the dispersion of any linear transformation of random variables using the known moments, making it essential for tasks such as error propagation and uncertainty quantification.

Covariance

Covariance is a measure of the joint variability of two random variables. It indicates the direction of the linear relationship between them; a positive covariance implies that the variables tend to increase or decrease together, while a negative one implies that one tends to increase when the other decreases. In the context of a joint pdf, the covariance is computed by taking the expected value of the product of the deviations of each variable from their respective means.

Joint Probability Density Function

A joint probability density function (pdf) describes the likelihood of two continuous random variables simultaneously taking on specific values. It provides information about the dependence between the variables and is defined over a specified support region where the density is positive. The joint pdf must integrate to one over this region, ensuring it conforms to the axioms of probability.

Expected Value

The expected value of a random variable (or a function of random variables) is a measure of its central tendency or mean. For functions of multiple random variables, the expectation is computed by integrating the product of the function and the joint pdf over the entire support. This concept is foundational for characterizing distributions and for performing operations such as linear transformations on random variables.

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