A random variable X has the following probability distribution: X… - Vidyadip

MCQ

A random variable $X$ has the following probability distribution:

$X = x$	1	2	3	4
P$(X = x)$	0.2	0.1	0.3	k

Then, the variance of X=

✓
1.29
B
1.31
C
1.27
D
1.23

✓

Answer

Correct option: A.

1.29

(A)
The sum of all the probabilities in a probability distribution is always unity.
$\begin{array}{ll}\therefore & 0.2+0.1+0.3+\mathrm{k}=1 \\\therefore & \mathrm{k}=1-0.6=0.4 \\& \mathrm{E}(\mathrm{X})=\sum x_{\mathrm{i}} \cdot \mathrm{P}\left(x_{\mathrm{i}}\right)\end{array}$
$\begin{aligned} & =1(0.2)+2(0.1)+3(0.3)+4(0.4) \\ & =0.2+0.2+0.9+1.6=2.9\end{aligned}$
$\begin{aligned} \operatorname{Var}(\mathrm{X}) & =\mathrm{E}\left(\mathrm{X}^2\right)-[\mathrm{E}(\mathrm{X})]^2 \\ & =(1)^2(0.2)+(2)^2(0.1)+(3)^2(0.3)+(4)^2(0.4)-(2.9)^2 \\ & =0.2+0.4+2.7+6.4-8.41 \\ & =9.7-8.41=1.29\end{aligned}$

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