Question

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

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

Answer 1

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

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

∴ k = \[\frac{1}{2}\]