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**Definition:** Let $X$ be a [continuous](/D/rvar-disc)[random variable](/D/rvar) with some [probability distribution](/D/dist) $P$ characterized by [probability density function](/P/pdf) $f_X(x)$. Then,
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* $P$ is called a unimodal probability distribution, if $f_X(x)$ has exactly one maximum;
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* $P$ is called a bimodal probability distribution, if $f_X(x)$ has exactly two maxima;
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* $P$ is called a trimodal probability distribution, if $f_X(x)$ has exactly three maxima;
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* $P$ is called a multimodal probability distribution, if $f_X(x)$ has more than one maximum.
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Note that this definition of multimodality differs from the [strict definition of the mode](/D/mode) in which only the global maximum of $f_X(x)$ [would be considered the single mode](/D/mode).
Then, any linear combination of $X$ and $Y$ follows a [univariate normal distribution](/D/norm):
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$$ \label{eq:bvn-lincomb}
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Z = a X + b Y \sim
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\mathcal{N}\left( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + 2ab \sigma_{12} + b^2 \sigma_2^2 \right) \; .
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$$
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**Proof:** The [linear transformation theorem for the multivariate normal distribution](/P/mvn-ltt) states that
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$$ \label{eq:mvn-ltt}
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X \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad Y = AX + c \sim \mathcal{N}(A\mu + c, A \Sigma A^\mathrm{T})
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$$
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where $X \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$ and $c \in \mathbb{R}^n$. In the present case, we have
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$$ \label{eq:X-A-a}
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X \in \mathbb{R}^2
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\quad \text{and} \quad
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A = \left[ \begin{matrix} a & b \end{matrix} \right] \in \mathbb{R}^{1 \times 2}
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\quad \text{and} \quad
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c = 0 \in \mathbb{R} \; ,
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$$
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such that
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$$ \label{eq:Z-X-Y}
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Z
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= A \left[ \begin{matrix} X \\ Y \end{matrix} \right] + c
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= \left[ \begin{matrix} a & b \end{matrix} \right] \left[ \begin{matrix} X \\ Y \end{matrix} \right] + 0
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= a X + b Y \; .
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$$
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Combining \eqref{eq:mvn-ltt}, \eqref{eq:bvn} and \eqref{eq:Z-X-Y}, it follows that
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$$ \label{eq:bvn-lincomb-qed}
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\begin{split}
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Z
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&\sim \mathcal{N}\left( \left[ \begin{matrix} a & b \end{matrix} \right] \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right], \left[ \begin{matrix} a & b \end{matrix} \right] \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \left[ \begin{matrix} a \\ b \end{matrix} \right] \right) \\
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&\sim \mathcal{N}\left( a \mu_1 + b \mu_2, \left[ \begin{matrix} a \sigma_1^2 + b \sigma_{12} & a \sigma_{12} + b \sigma_2^2 \end{matrix} \right] \left[ \begin{matrix} a \\ b \end{matrix} \right] \right) \\
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&\sim \mathcal{N}\left( a \mu_1 + b \mu_2, (a^2 \sigma_1^2 + ab \sigma_{12}) + (ab \sigma_{12} + b^2 \sigma_2^2) \right) \\
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&\sim \mathcal{N}\left( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + 2ab \sigma_{12} + b^2 \sigma_2^2 \right) \; .
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