Consider the probability distribution of a random variable X: X 0… - Vidyadip

Question

Consider the probability distribution of a random variable X:

X	0	1	2	3	4
P(X)	0.1	0.25	0.3	0.2	0.15

Calculate:

$\text{V}\Big(\frac{\text{X}}{2}\Big)$
Variance of X.

✓

Answer

We have,

$X$	0	1	2	3	4
$P(X)$	0.1	0.25	0.3	0.2	0.15
$XP(X)$	0	0.25	0.6	0.6	0.60
$X^2P(X)$	0	0.25	1.2	1.8	2.40

var$(\text{X})=\text{E}(\text{X}^2)-\big[\text{E}(\text{X})\big]^2$
Where, $\text{E}(\text{X})=\mu=\sum_\limits{\text{i}=1}^\text{n}\text{x}^2_1\text{P}(\text{x}_1)$
$\therefore\text{E}(\text{X})=0+0.25+0.6+0.6+0.60=2.05$
And $\text{E}(\text{X}^2)=0+0.25+1.2+0.8+2.40=5.65$

$\text{V}\Big[\frac{\text{X}}{2}\Big]=\frac{1}{4}\text{V}(\text{X})$

$=\frac{1}{4}\big[5.65-(2.05^2)\big]$

$=\frac{1}{4}[5.65-4.2025]$

$=\frac{1}{4}\times1.4475$

$=0.361875$

$\text{V}(\text{X})=1.4475$

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