Question

A r.v. $X$ has the following probability distribution:

$X=x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$P\left(x\right)$	$0.1$	$k$	$0.2$	$2k$	$0.3$	$k$

Find the value of $k$ and calculate mean and variance of $X$.

Answer 1

$x=x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$P\left(x\right)$	$0.1$	$k$	$0.2$	$2k$	$0.3$	$k$

Since sum of probabilities $=1$

$0.1+k+0.2+2k+0.3+k=1$

$4k=1-0.6$

$k=0.1$

Mean $=(-2)0.1+(-1)k+\left(0.2\right)\left(o\right)+1\left(2k\right)+2\left(0.3\right)+3\left(k\right)$

$=4k+0.4$

$=4\left(0.1\right)+0.4$

$\left(\mu \right)=0.8$

Variance $\Rightarrow \left(6^2\right)=\sum _{i=1}^n{x}^2\left(p\right(x=xi)-\mu i$

$6^2=4\left(0.1\right)+1\left(k\right)+0\left(0.2\right)+1\left(2k\right)+4\left(0.3\right)+9k-{\mu }^2$

$=12\left(0.1\right)+1.6-(0.8{)}^2$

$=1.2+1.6-0.64$

$6^2\Rightarrow 2.16$