the table below shows the probability distribution of a random variable z.\n|z|p(z)|\n|-2|0.09|\n|-1|0.13|\n|…

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:

$1.98$