The random variable $X$ taken on the values 1, 2 or 3 with probabilities $\dfrac{2 + 5p}{5}$, $\dfrac{1 + 3p}{5}$, $\dfrac{1.5 + 2p}{5}$ respectively. Find the values of $p$ and $E(X)$.

The random variable $X$ taken on the values 1, 2 or 3 with probabilities $\dfrac{2 + 5p}{5}$, $\dfrac{1 + 3p}{5}$, $\dfrac{1.5 + 2p}{5}$ respectively. The values of $p$ and $E(X)$ are respectively 

(A) 0.05, 1.87 

(B) 1.90, 5.87 

(C) 0.05, 1.10 

(D) 0.25, 1.40 

(GATE 2007) 

Answer: (A) 0.05, 1.87 

Explaination: 

$\dfrac{2 + 5p}{5} + \dfrac{1 + 3p}{5} + \dfrac{1.5 + 2p}{5} = 1$ 

$\Rightarrow  4.5 + 10p = 5$ 

$\Rightarrow p = 0.05$ 

Now, $E(X)$ 

$= \sum x \times P(X = x)$ 

$= 1 \times \left(\dfrac{2 + 5p}{5}\right) + 2 \times \left(\dfrac{1 + 3p}{5}\right) + 3 \times \left(\dfrac{1.5 + 2p}{5}\right)$ 

$= \dfrac{8.5 + 17p}{5}$ 

$= \dfrac{8.5 + 0.85}{5}$ 

$= \dfrac{9.35}{5}$ 

$= 1.87$