Question

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

x	p(x)
-8	0.56
-7	0.3
-6	0.14

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

Explanation:

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

$E(X)=\sum_{i}x_ip_i=(-8)\times0.56+(-7)\times0.3+(-6)\times0.14=-7.46$

Step2: Calculate the variance $Var(X)$

$Var(X)=\sum_{i}(x_i - E(X))^{2}p_i$
$( - 8-(-7.46))^{2}\times0.56+( - 7-(-7.46))^{2}\times0.3+( - 6-(-7.46))^{2}\times0.14$
$=(-0.54)^{2}\times0.56+(0.46)^{2}\times0.3+(1.46)^{2}\times0.14$
$=0.2916\times0.56 + 0.2116\times0.3+2.1316\times0.14$
$=0.163296+0.06348+0.298424 = 0.5252$

Step3: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{Var(X)}=\sqrt{0.5252}\approx0.72$

Answer:

$0.72$