Q.
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
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | a | a | a | b | b | 0.3 |
TS EAMCET 2017
Solution:
We have
x
$P(x)$
$P_{i}x_{i}$
1
a
a
2
a
2a
3
a
3a
4
b
4b
5
b
5b
6
0.3
1.8
$\sum P_{i} =a+a+a+b+b+0.3 $
$\Rightarrow 1 =3 \,a+2 \,b+0.3$
$\Rightarrow 3 a+2 b =0.7 \ldots(i) $
$\Sigma P_{i} x_{i} =a+2\, a+3 \,a+4 \,b+5 \,b+1.8 $
$\Rightarrow 4.2 =6 \,a+9\, b+1.8$
$\Rightarrow 2 \,a+3 \,b =0.8 \ldots (ii)$
Solving Eqs. (i) and (ii), we get
$a=0.1, b=0.2$
| x | $P(x)$ | $P_{i}x_{i}$ |
|---|---|---|
| 1 | a | a |
| 2 | a | 2a |
| 3 | a | 3a |
| 4 | b | 4b |
| 5 | b | 5b |
| 6 | 0.3 | 1.8 |