Question

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

y	p(y)
-13	0.22
-12	0.4
-11	0.21
-10	0.17

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

Explanation:

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

$E(Y)=\sum_{i}y_iP(y_i)=(-13)\times0.22+(-12)\times0.4+(-11)\times0.21+(-10)\times0.17=-12.03$

Step2: Calculate the variance $Var(Y)$

$Var(Y)=\sum_{i}(y_i - E(Y))^{2}P(y_i)$
$=((-13)-(-12.03))^{2}\times0.22+((-12)-(-12.03))^{2}\times0.4+((-11)-(-12.03))^{2}\times0.21+((-10)-(-12.03))^{2}\times0.17$
$=(-0.97)^{2}\times0.22+(0.03)^{2}\times0.4+(1.03)^{2}\times0.21+(2.03)^{2}\times0.17$
$=0.9409\times0.22 + 0.0009\times0.4+1.0609\times0.21+4.1209\times0.17$
$=0.2070+0.0004+0.2228+0.7006$
$=1.1308$

Step3: Calculate the standard deviation $\sigma_Y$

$\sigma_Y=\sqrt{Var(Y)}=\sqrt{1.1308}\approx1.06$

Answer:

$1.06$