Question

A discrete random variable X has the probability distribution given below:

X:	0.5	1	1.5	2
P(X):	k	k2	2k2	k

Find the value of k.

Answer 1

The probability distribution of X is given as:
 

X:	0.5	1	1.5	2
P(X):	k	k2	2k2	k

\[ \text{ As}, \sum p_i = 1\]
\[ \Rightarrow k + k^2 + 2 k^2 + k = 1\]
\[ \Rightarrow 3 k^2 + 2k - 1 = 0\]
\[ \Rightarrow 3 k^2 + 3k - k - 1 = 0\]
\[ \Rightarrow 3k\left( k + 1 \right) - 1\left( k + 1 \right) = 0\]
\[ \Rightarrow \left( 3k - 1 \right)\left( k + 1 \right) = 0\]
\[ \Rightarrow k = \frac{1}{3} or k = - 1\]
\[\text{ but k cannot be negative } \]
\[ \therefore k = \frac{1}{3}\]