$X = x$ $- 2$ $- 1$ $0$ $1$ $2$ $3$
$P (X = x)$ $0.1$ $K$ $0.2$ $2 K$ $0.3$ $K$
is the probability distribution of random variable $X$ . Find the value of $K$ and variance of $X$ .

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Solution

$\begin{matrix} \sum P (X = x) = 1 \Rightarrow 0.1 + k + 0.2 + 2 k + 0.3 + k = 1 \Rightarrow k = 0.1 \end{matrix}$
Therefore,
$X = x$ $- 2$ $- 1$ $0$ $1$ $2$ $3$
$P (X = x)$ $0.1$ $0.1$ $0.2$ $0.2$ $0.3$ $0.1$
Variance ( $X$ ) $= {(x - μ)}^{2} P (X = x)$
$\begin{matrix} μ = - 2 \times 0.1 + (- 1) (0.1) + 0 (0.2) + 1 (0.2) + 2 (0.3) + 3 (0.1) μ = 0.8 \end{matrix}$
Therefore,
Var( $X$ ) $= {(- 2 - 0.8)}^{2} (0.1) + {(- 1 - 0.8)}^{2} (0.1) + {(0 - 0.8)}^{2} (0.2) + {(1 - 0.8)}^{2} (0.2) + {(2 - 0.8)}^{2} (0.3) + {(3 - 0.8)}^{2} (0.1) = 2.05$

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