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X is a random variable with distribution given below
$X :$ 0 1 2 3
$P (x = x) :$ k 3k 3k k
then the value of k and variance are

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$\frac{1}{8}, \frac{22}{27}$

$\frac{1}{8}, \frac{23}{27}$

$\frac{1}{8}, \frac{24}{27}$

$\frac{1}{8}, \frac{3}{4}$

answer is D.

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Detailed Solution

$Σ P (X = r) = 1$

$\Rightarrow 8 k = 1 \Rightarrow k = \frac{1}{8}$

$μ = Σ x_{i} P (X = X_{i}) = 12 k = \frac{12}{8} = \frac{3}{2}$

$σ^{2} + μ^{2} = 3 k + 12 k + 9 k = 24 k = 3$

$σ^{2} = 3 - \frac{9}{4} = \frac{3}{4}$

k= $\frac{1}{8}$ and Variance = $\frac{3}{4}$

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