Question

Consider the following functions of the random variables $Y_{1}, Y_{2},$ and $Y_{3}$ $$\begin{array}{l} W_{1}=Y_{1}+Y_{2}+Y_{3} \\ W_{2}=Y_{1}-Y_{2} \\ W_{3}=Y_{1}-Y_{2}-Y_{3} \end{array}$$ a. State the above in matrix notation. b. Find the expectation of the random vector $\mathbf{W}$. c. Find the variance-covariance matrix of $\mathbf{W}$.

          Consider the following functions of the random variables $Y_{1}, Y_{2},$ and $Y_{3}$
$$\begin{array}{l}
W_{1}=Y_{1}+Y_{2}+Y_{3} \\
W_{2}=Y_{1}-Y_{2} \\
W_{3}=Y_{1}-Y_{2}-Y_{3}
\end{array}$$
a. State the above in matrix notation.
b. Find the expectation of the random vector $\mathbf{W}$.
c. Find the variance-covariance matrix of $\mathbf{W}$.

Added by Todd C.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Supreeta N

03/08/2023

Step 1

Notice that the given relationships can be written as linear combinations of Y₁, Y₂, and Y₃. In particular, W₁ = 1·Y₁ + 1·Y₂ + 1·Y₃, W₂ = 1·Y₁ + (–1)·Y₂ + 0·Y₃, W₃ = 1·Y₁ + (–1)·Y₂ + (–1)·Y₃. Thus the transformation matrix A that relates W and Y is A = [ Show more…

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Consider the following functions of the random variables $Y_{1}, Y_{2},$ and $Y_{3}$ $$\begin{array}{l} W_{1}=Y_{1}+Y_{2}+Y_{3} \\ W_{2}=Y_{1}-Y_{2} \\ W_{3}=Y_{1}-Y_{2}-Y_{3} \end{array}$$ a. State the above in matrix notation. b. Find the expectation of the random vector $\mathbf{W}$. c. Find the variance-covariance matrix of $\mathbf{W}$.