There 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 μ and σ 2 rep

Question

There 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 μ and σ 2 represent mean and variance of X, respectively, then 10 μ 2 + σ 2 is equal to

TestHub · Accepted Answer

There rotten apples are mixed accidently with seven good apples,

And four apples are drawn one by one without replacement.

Now the random variable X denote the number of rotten apples.

And $μ$ and $σ^{2}$ represent mean and variance of X, respectively,

So, now plotting the table of given data we get,

$x$	$P (x)$	$x P (X)$	$x^{2} P (X)$
$0$	$\frac{1}{6}$	$0$	$0$
$1$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
$2$	$\frac{3}{10}$	$\frac{6}{10}$	$\frac{12}{10}$
$3$	$\frac{1}{30}$	$\frac{1}{10}$	$\frac{9}{30}$

Now mean is given by, $\sum x P (x) = \frac{6}{2} = μ$

And variance is given by $σ^{2} = \sum x^{2} P (x) - μ^{2}$

So, by putting the value from above table we get, $σ^{2} + μ^{2} = 0 + \frac{1}{2} + \frac{12}{10} + \frac{9}{30} = 2$

Hence, $10 (σ^{2} + μ^{2}) = 20$

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