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The probability distribution for the number of girls among 8 children is provided.

The variable x is the number of girls among 8 children.

The requirements are as follows:

1)The variable x is anumerical random variable.

2)The sum of the probabilities is computed as:

$$\begin{gathered} 鈭�P\left(x\right)=0.004+0.031+0.109+...+0.004 \\ =0.999 \end{gathered}$$

Therefore,the sum of the probabilities is approximately equal to 1 with a round of error as 0.001.

3) Each value of P(x) is between 0 and 1.

Thus, there are no requirements that are not satisfied.

The mean for a random variable is computed as:

$$\begin{gathered} 渭=鈭�\left[x脳P\left(x\right)\right] \\ =\left(0脳0.004\right)+\left(1脳0.031\right)+\left(2脳0.109\right)+...+\left(8脳0.004\right) \\ =3.996 \\ 鈮�4 \end{gathered}$$

Thus, the mean number of girls is 4.

The standard deviation of the random variable x is computed as:

$蟽=\sqrt{鈭�\left[x^2脳P\left(x\right)\right]-{渭}^2}$

The calculations are as follows:

$$\begin{gathered} 鈭�\left[x^2路P\left(x\right)\right]=\left(0^2脳0.004\right)+\left(1^2脳0.031\right)+\left(2^2脳0.109\right)+...+\left(8^2脳0.004\right) \\ =17.98 \end{gathered}$$

The standard deviation is given as:

$$\begin{gathered} 蟽=\sqrt{鈭�\left[x^2路P\left(x\right)\right]-{渭}^2} \\ =\sqrt{17.98-{3.996}^2} \\ =1.41 \\ 鈮�1.4 \end{gathered}$$

Thus, the standard deviation of x is 1.4.

Significant high values are the values greater than or equal to $渭+2蟽$ .

The calculations are computed as:

$$\begin{gathered} 渭+2蟽=4+\left(2脳1.4\right) \\ =6.8 \end{gathered}$$

Since 6 girls are less than 6.8 girls, this implies that it is not a significantly high number of girls.