the table below shows the probability distribution of a random variable y.\n|y|p(y)|\n|5|0.4|\n|6|0|\n|7|0.4|…

Explanation:

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

$E(Y)=\sum_{i}y_ip(y_i)=5\times0.4 + 6\times0+7\times0.4 + 8\times0.2=2 + 0+2.8 + 1.6 = 6.4$

Step2: Calculate the variance $Var(Y)$

$Var(Y)=\sum_{i}(y_i - E(Y))^{2}p(y_i)=(5 - 6.4)^{2}\times0.4+(6 - 6.4)^{2}\times0+(7 - 6.4)^{2}\times0.4+(8 - 6.4)^{2}\times0.2$ $=(-1.4)^{2}\times0.4+( - 0.4)^{2}\times0+(0.6)^{2}\times0.4+(1.6)^{2}\times0.2$ $=1.96\times0.4+0 + 0.36\times0.4+2.56\times0.2$ $=0.784+0 + 0.144+0.512 = 1.44$

Step3: Calculate the standard deviation $\sigma_Y$

$\sigma_Y=\sqrt{Var(Y)}=\sqrt{1.44}=1.20$

Answer:

$1.20$