For what value of k the following distribution is a probability distribution?

X = x_i : 0 1 2 3

P (X = x_i) : 2k⁴ 3k² − 5k³ 2k − 3k² 3k − 1

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Solution

We know that the sum of probabilities in a probability distribution is always 1.

∴ P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3) = 1

$\Rightarrow 2 k^{4} + 3 k^{2} - 5 k^{3} + 2 k - 3 k^{2} + 3 k - 1 = 1 \Rightarrow 2 k^{4} - 5 k^{3} + 5 k = 2 \Rightarrow 2 k^{4} - 5 k^{3} + 5 k - 2 = 0 \Rightarrow (k - 1) (k - 2) (2 k^{2} + k - 1) = 0 \Rightarrow (k - 1) (k - 2) (2 k - 1) (k + 1) = 0 \Rightarrow k = - 1, \frac{1}{2}, 1, 2 (Neglecting - 1, 1 and 2 as they give the value of probability negative or greater than 1)$

∴ k = $\frac{1}{2}$

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X = x_i :	0	1	2	3
P (X = x_i) :	2k⁴	3k² − 5k³	2k − 3k²	3k − 1