Question

The probability distribution of random variable X is given by:

X	1	2	3	4	5
P(X)	K	2K	2K	3K	K

Let $p = P (1< X <4 \mid X <3)$. If $5 p =\lambda K$, then $\lambda$ equal to_____

Answer 1

$\sum P ( X )=1 \Rightarrow k +2 k +2 k +3 k + k =1$
$\Rightarrow k =\frac{1}{9}$
Now, $p = P \left(\frac{ kX < 4}{ X < 3}\right)=\frac{ P ( X = 2)}{ P ( X < 3)}$
$=\frac{\frac{2 k }{9 k }}{\frac{ k }{9 k }+\frac{2 k }{9 k }}=\frac{2}{3}$
$\Rightarrow p =\frac{2}{3}$
Now, $5 p =\lambda k$
$\Rightarrow (5)\left(\frac{2}{3}\right)=\lambda(1 / 9)$
$\Rightarrow \lambda=30$