Expectation of the product of two matrices

Question

I recently viewed this question and I decided it to give it a try. I got stuck in the computation of a certain expectation and variance. Summarizing the problem, this is the issue:

Let $X\in R^{n\times d}$ be a data matrix, $R\in R^{n\times d}$ a matrix such that $ R_{ij}\sim\text{Bern}(p)$ is sampled i.i.d. from the Bernoulli distribution, $M=X\odot R$ the element-wise product of the previous two matrices, and $\Gamma$ be a diagonal matrix with $\Gamma_{ii}=(X^\top X)_{ii}^{1/2}$.

Show that $E[M]=pX$, and $var(M)=p(1-p)\Gamma²$

I first went considering both the columns of $X$ and $R$ as independent vectors. Using a specific example where both $X$ and $R$ have three columns, i.e. $X=[A\ B\ C]^T$, and $R=[\alpha\ \beta\ \delta]^T$, we have that $M=[\alpha A\ \beta B\ \delta C]^T$. Taking the expectation of this: $$E[M]= \begin{bmatrix} E[\alpha A] \\ E[\beta B]\\ E[\delta C] \\ \end{bmatrix} \stackrel{\star}{=} \begin{bmatrix} E[\alpha]E[A] \\ E[\beta]E[B]\\ E[\delta]E[C] \\ \end{bmatrix} = \begin{bmatrix} pE[A] \\ pE[B]\\ pE[C] \\ \end{bmatrix}=pE[X] $$

where I used that $\star$ the variables are independent.

It doesn't look like the answer is correct and I am not sure what I am misunderstanding. For the variance, I don't even know where to start.

Answer 1

I just realised what is the answer (and also thanks to the reflection questions of @Gateau-Gallois).

1. Mean: $X$ is taken as a matrix of constants whereas $R\sim Bern(p)$. That's why $E[M]=pX$.

2. Variance: Following a similar argument, one can see then that the covariance matrix is: $$ \begin{align} var(M_{ij})&=var(X_{ij}R_{ij})\stackrel{\star}{=}X_{ij}^2var(R_{ij}) \end{align}$$ where $\star$ is the same assumption as in the question and the fact that $var(aX)=a^2var(X)$. The expression results in 0 if $i\neq j$ since $R_{ij}$ are i.i.d. and otherwise is $X^2_{ii}var(R_{ii})=X^2_{ii}p(1-p)$. Taking, then, the matrix form: $$ \begin{align} var(M)=p(1-p)(X^TX)_{ii}\stackrel{\star^2}{=}p(1-p)\Gamma^2 \end{align} $$ where $\star^2$ is the definition $\Gamma=(X^TX)_{ii}^{1/2}$stated in the question.