A random variable $^{'} X^{'}$ has the following probability distribution:

$X$ $1$ $2$ $3$ $4$ $5$ $6$ $7$

$P (X)$ $k - 1$ $3 k$ $k$ $3 k$ $3 k^{2}$ $k^{2}$ $k^{2} + k$

Then the value of $k$ is

\frac{2}{7}

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\frac{1}{5}

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\frac{1}{10}

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- 2

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Solution

The correct option is B $\frac{1}{5}$
$\sum P_{i} = 1$
$\Rightarrow k - 1 + 3 k + k + 3 k + 3 k^{2} + k^{2} + k^{2} + k = 1$
$5 k^{2} + 9 k - 2 = 0 \Rightarrow (5 k - 1) (k + 2) = 0$
$∴ k = \frac{1}{5}$ $[∵ k = - 2$ is not possible $]$ .

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$X$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P (X)$	$k - 1$	$3 k$	$k$	$3 k$	$3 k^{2}$	$k^{2}$	$k^{2} + k$