Let $X$ and $Y$ be two Bernoulli random variables, having parameters $\lambda \in ]0,1[$ and $\mu \in ]0,1[$, respectively. Calculate $d_{VT}\left(p_X, p_Y\right)$.

If $x$ and $y$ are two probability distributions on $\mathbf{N}$, we define the application $x * y : \mathbf{N} \rightarrow \mathbf{R}_+$ by $$\forall k \in \mathbf{N} \quad (x * y)(k) = \sum_{i=0}^{k} x(i) y(k-i) = \sum_{i+j=k} x(i) y(j)$$ Show that $x * y$ is a distribution on $\mathbf{N}$.

Let $\mathrm { A } = \left[ \mathrm { a } _ { i j } \right]$ be a $2 \times 2$ matrix such that $\mathrm { a } _ { i j } \in \{ 0,1 \}$ for all $i$ and $j$. Let the random variable X denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is :
(1) $\frac { 3 } { 4 }$
(2) $\frac { 5 } { 8 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 4 }$