Q. Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable $X$ denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent mean and variance of $X$, respectively, then $10\left(\mu^2+\sigma^2\right)$ is equal to
Solution:
$X$
$P(X)$
$XP(X)$
$X^2P(X)$
0
1/6
0
0
1
1/2
1/2
1/2
2
3/10
6/10
12/10
4
1/30
1/10
9/30
$ \sum xP ( x )=\frac{6}{2}=\mu $
$ \sigma^2=\sum x ^2 P ( x )-\mu^2 $
$ \sigma^2+\mu^2=0+\frac{1}{2}+\frac{12}{10}+\frac{9}{30}=2$
$ 10\left(\sigma^2+\mu^2\right)=20 \text { Ans. }$
| $X$ | $P(X)$ | $XP(X)$ | $X^2P(X)$ |
|---|---|---|---|
| 0 | 1/6 | 0 | 0 |
| 1 | 1/2 | 1/2 | 1/2 |
| 2 | 3/10 | 6/10 | 12/10 |
| 4 | 1/30 | 1/10 | 9/30 |