For the following probability distribution determine standard dev… - Vidyadip

Question

For the following probability distribution determine standard deviation of the
random variable X.

X	2	3	4
P(X)	0.2	0.5	0.3

✓

Answer

We have

$X$	2	3	4
$P(X)$	0.2	0.5	0.3
$XP(X)$	0.4	1.5	1.2
$X^2P(X)$	0.8	4.5	4.8

We know that, standard deviation of $\text{X}=\sqrt{\text{Var}(\text{X})}$
Where, $\text{var}(\text{X})=\text{E}(\text{X}^2)-\big[\text{E}(\text{X})\big]^2$
$=\sum_\limits{\text{i}=1}^\text{n}\text{x}^2_1\text{P}(\text{X}_1)-\Bigg[\sum_\limits{\text{i}=1}^\text{n}\text{x}_\text{i}\text{P}_{\text{i}}^2\Bigg]$
$\therefore\text{Var}(\text{X})=[0.8+4.5+4.8]-[0.4+1.5+1.2]^2$
$=10.1-(3.1)^2=10.1-9.61$
$=0.49$
$\therefore$ Standard deviation of $\text{X}=\sqrt{\text{Var}\text{X}}$
$=\sqrt{0.49}=0.7$

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