Question

the table below shows the probability distribution of a random variable y.
y p(y)
-14 0.03
-13 0.02
-12 0.35
-11 0.1
-10 0.5
what is the standard deviation of y?
round your answer to the nearest hundredth.

Explanation:

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

$E(Y)=\sum_{i}y_iP(y_i)=(-14)\times0.03+(-13)\times0.02+(-12)\times0.35+(-11)\times0.1+(-10)\times0.5$
$=- 0.42-0.26 - 4.2-1.1 - 5=-10.98$

Step2: Calculate the variance $Var(Y)$

$Var(Y)=\sum_{i}(y_i - E(Y))^{2}P(y_i)$
$=((-14 + 10.98)^{2}\times0.03+(-13 + 10.98)^{2}\times0.02+(-12 + 10.98)^{2}\times0.35+(-11 + 10.98)^{2}\times0.1+(-10 + 10.98)^{2}\times0.5)$
$=( (-3.02)^{2}\times0.03+(-2.02)^{2}\times0.02+( - 1.02)^{2}\times0.35+( - 0.02)^{2}\times0.1+(0.98)^{2}\times0.5)$
$=(9.1204\times0.03 + 4.0804\times0.02+1.0404\times0.35 + 0.0004\times0.1+0.9604\times0.5)$
$=(0.273612+0.081608 + 0.36414+0.00004+0.4802)$
$=1.1996$

Step3: Calculate the standard deviation $\sigma(Y)$

$\sigma(Y)=\sqrt{Var(Y)}=\sqrt{1.1996}\approx1.10$

Answer:

$1.10$