Assume that Y1, Y2, and Y3 are random variables, with E(Y1)=2, E(Y2)=-1, E(Y3)=4 V(Y1)=4,

step 1 So in this question we have that the expected value of y1 is 2, for y2 that expected value is mi

Assume that Y1, Y2, and Y3 are random variables, with...

Assume that $Y_{1}, Y_{2},$ and $Y_{3}$ are random variables, with $$E\left(Y_{1}\right)=2, \quad E\left(Y_{2}\right)=-1, \quad E\left(Y_{3}\right)=4 $$ $$V\left(Y_{1}\right)=4, \quad V\left(Y_{2}\right)=6, \quad V\left(Y_{3}\right)=8$$ $$\operatorname{Cov}\left(Y_{1}, Y_{2}\right)=1, \operatorname{Cov}\left(Y_{1}, Y_{3}\right)=-1, \operatorname{Cov}\left(Y_{2}, Y_{3}\right)=0$$ Find $E\left(3 Y_{1}+4 Y_{2}-6 Y_{3}\right)$ and $V\left(3 Y_{1}+4 Y_{2}-6 Y_{3}\right)$

Assume that $Y_{1}, Y_{2},$ and $Y_{3}$ are random variables, with
$$E\left(Y_{1}\right)=2, \quad E\left(Y_{2}\right)=-1, \quad E\left(Y_{3}\right)=4 $$
$$V\left(Y_{1}\right)=4, \quad V\left(Y_{2}\right)=6, \quad V\left(Y_{3}\right)=8$$
$$\operatorname{Cov}\left(Y_{1}, Y_{2}\right)=1, \operatorname{Cov}\left(Y_{1}, Y_{3}\right)=-1, \operatorname{Cov}\left(Y_{2}, Y_{3}\right)=0$$
Find $E\left(3 Y_{1}+4 Y_{2}-6 Y_{3}\right)$ and $V\left(3 Y_{1}+4 Y_{2}-6 Y_{3}\right)$

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Key Concepts

Covariance

Covariance quantifies the extent to which two random variables change together. It plays a crucial role in determining the variance of a sum or difference of random variables, as it captures the dependency between variables that can either amplify or diminish the overall variability of the combined expression.

Linearity of Expectation

This principle states that the expected value of a linear combination of random variables is equal to the linear combination of their individual expected values. Importantly, this property holds regardless of whether the random variables are independent, making it a foundational concept in probability and statistics.

Variance of Linear Combinations

When calculating the variance of a linear combination of random variables, one must account for both the individual variances and the covariances between the variables. The contribution of each variable's variance is weighted by the square of its coefficient, and the covariances between different variables are incorporated with products of their respective coefficients.

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