Question

the table below shows the probability distribution of a random variable x.
x p(x)
-6 0.17
-5 0.69
-4 0.06
-3 0.08
what is the standard deviation of x?
round your answer to the nearest hundredth.

Explanation:

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

$E(X)=\sum_{i}x_iP(x_i)=(-6)\times0.17+(-5)\times0.69+(-4)\times0.06+(-3)\times0.08$
$=- 1.02-3.45 - 0.24-0.24=-4.95$

Step2: Calculate the variance $Var(X)$

$Var(X)=\sum_{i}(x_i - E(X))^{2}P(x_i)$
For $x_1=-6$: $(-6 - (-4.95))^{2}\times0.17=(-1.05)^{2}\times0.17 = 1.1025\times0.17=0.187425$
For $x_2=-5$: $(-5 - (-4.95))^{2}\times0.69=(-0.05)^{2}\times0.69 = 0.0025\times0.69 = 0.001725$
For $x_3=-4$: $(-4 - (-4.95))^{2}\times0.06=(0.95)^{2}\times0.06=0.9025\times0.06 = 0.05415$
For $x_4=-3$: $(-3 - (-4.95))^{2}\times0.08=(1.95)^{2}\times0.08 = 3.8025\times0.08=0.3042$
$Var(X)=0.187425 + 0.001725+0.05415+0.3042=0.5475$

Step3: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{Var(X)}=\sqrt{0.5475}\approx0.74$

Answer:

$0.74$