The probability distribution of a random variable $X$ is given below:
$x$ $1$ $2$ $3$ $4$ $5$ $6$
$P (X = x)$ $a$ $a$ $a$ $b$ $b$ $0.3$
If mean of $X$ is $4.2$ , then $a$ and $b$ are respectively equal to

0.3, 0.2

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0.1, 0.4

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0.1, 0.2

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0.2, 0.1

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Solution

The correct option is A $0.1, 0.2$
Given, mean of $x$ is $4.2$
Thus that $a + 2 a + 3 a + 4 b + 5 b + 1.8 = 4.2$
$\Rightarrow 6 a + 9 b = 2.4$ .....(1)
Also $a + a + a + b + b + 0.3 = 1$
$\Rightarrow 3 a + 2 b = 0.7$ .....(2)
Multiply equation (2) by $2$ ,
$6 a + 4 b = 1.4$ .....(3)
Subtracting equations (1) and (3), we get
$5 b = 1.0 \Rightarrow b = 0.2$
Substituting this value in equation (1), we get
$6 a + 9 (0.2) = 2.4$
$\Rightarrow 6 a + 1.8 = 2.4 \Rightarrow a = 0.1$
Therefore, $a = 0.1, b = 0.2$

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