Question

the table below shows the probability distribution of a random variable z.

z	p(z)
-2	0.09
-1	0.13
0	0.13
1	0.03
2	0.04
3	0.48
4	0.1

what is the standard deviation of z?
round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the expected value $E(Z)$

$E(Z)=\sum_{i}z_iP(z_i)=(- 2)\times0.09+(-1)\times0.13 + 0\times0.13+1\times0.03 + 2\times0.04+3\times0.48+4\times0.1$
$=-0.18 - 0.13+0 + 0.03+0.08+1.44+0.4$
$=1.64$

Step2: Calculate $E(Z^{2})$

$E(Z^{2})=\sum_{i}z_{i}^{2}P(z_i)=(-2)^{2}\times0.09+(-1)^{2}\times0.13 + 0^{2}\times0.13+1^{2}\times0.03 + 2^{2}\times0.04+3^{2}\times0.48+4^{2}\times0.1$
$=4\times0.09 + 1\times0.13+0+1\times0.03+4\times0.04+9\times0.48+16\times0.1$
$=0.36+0.13 + 0+0.03+0.16+4.32+1.6$
$=6.6$

Step3: Calculate the variance $Var(Z)$

$Var(Z)=E(Z^{2})-[E(Z)]^{2}=6.6-(1.64)^{2}$
$=6.6 - 2.6896$
$=3.9104$

Step4: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{Var(Z)}=\sqrt{3.9104}\approx1.98$

Answer:

$1.98$