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Suppose we have $p$ random variates :- $X_1, \cdots , X_p$. $n$ observations involving each random variable are made . Thus, the $jth $ trial or $jth$ observation is a vector of the form : $[x_{j1},x_{j2},\cdots, x_{jp}]^T$

where $x_{ji}$ means the value in the $jth$ trial for the $ith $ random variable.

Let matrix $X = [X_{.,1} ~~ X_{.,2} ~~ \cdots X_{.,p}]_{n \times p}$

Now, if we wish to investigate the linear combination of the $p$ variates, I believe the expression should be

$$c^T_{1 \times p} ~X^T_{p \times n} = c_1 X_1 + \cdots c_pX_p~~ $$

where $X_i$ is the $ith$ row of $X$ and represents the $ith$ trial.

But, my textbook multivariate statistical analysis by Richard Johnson seems to have a typo here in the image attached below. Could Someone please confirm the same? Could someone confirm the same.

MathMan
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  • It seems to me like the book is using bold for vectors and normal weight font for scalars, so I suspect that in $c'X =c_1X_1 +\ldots$ the $X$ is meant to be a random vector and the $X_i$ are random variables. $X_i$ definitely doesn't represent the $i^\mathrm{th}$ trial, because equation (3-31) shows that the linear combination is of the variables, not the trials. $c_i$ is multiplied by $x_{ji}$, where $j$ represents the trial and $i$ represents the variable. – Joe Sep 26 '21 at 11:35
  • @Joe yes the book is using bold for vectors. But, isn't still there an error? The matrix dimensions don't comply for multiplication! – MathMan Sep 26 '21 at 11:42
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    I'm saying that I don't think $\mathbf{X}$ is meant to represent a matrix. But you are correct that if we represent all of the trials in a $n \times p$ matrix, then the $n$ trials of the linear combination would be given by $\mathbf{X}\mathbf{c}= c_1\mathbf{X}_1+\ldots+c_p\mathbf{X}_p$. However, $\mathbf{X}_i$ would be the $i^\mathrm{th}$ column of $\mathbf{X}$, representing the $i^\mathrm{th}$ variable across all trials. – Joe Sep 26 '21 at 12:34

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