Question

Let $X$ be the discrete random variable representing the number $(x)$ appeared on the face of a biased die when it is rolled. The probability distribution of $X$ is

$X=X$	1	2	3	4	5	6
$P(X=x)$	0.1	0.15	0.3	0.25	k	k

• A. 1.64
• B. 1.93
• C. 2.16
• D. 2.28

Answer 1

Answer: (B)

We know that,
$\Sigma P(X)=1$
$\Rightarrow 0.1+0.15+0.3+0.25+k+k=1$
$\Rightarrow 0.80+2 k=1$
$\Rightarrow 2 k=0.2 $
$\Rightarrow k=0.1$
Now, mean, $\bar{X}=\Sigma X P(X)$
$=1 \times 0.1+2 \times 0.15+3 \times 0.3+4 \times 0.25+5 k+6 k$
$=3.4[K=0.1]$
and
$\Sigma X^{2} P(X)=1 \times 0.1+4 \times 0.15+9 \times 0.3$
$+16 \times 0.25+25 k+36 k=13.5$
$\therefore $ Variance $=\Sigma X^{2} P(X)-(\Sigma X P(X))^{2}$
$=13.5-(3.4)^{2}=13.5-11.56$
$=1.94$ or $1.93$