Question

(5.1.15) Refer to the accompanying​ table, which describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children. Find the mean and standard deviation for the number of girls in 8 births.
Number of
Girls x ​P(x)
0 0.006
1 0.035
2 0.109
3 0.222
4 0.277
5 0.213
6 0.107
7 0.026
8 0.005
The mean is mu ? ​girl(s)

Answer 1

The following table shows the provided outputs of the discrete random variables, along with the corresponding probabilities:

Therefore, the population mean is calculated as follows:

$\mu =\sum_{i=1}^{n} x_{i}*P(x_i)$

=0*0.006+1*0.035+2*0.109+3*0.222+4*0.277+5*0.213+6*0.107+7*.026+8*0.005

=3.956

Standard deviation:

We first compute the expected value of $X^2$ .

$E(X^2)=\sum_{i=1}^{n} x_{i}^2*P(x_i^2)$

=$0 2 *0.006+1 2 *0.035+2 2 *0.109+3 2 *0.222+4 2 *0.277+5 2 *0.213+6 2 *0.107+7 2 *.026+8 2 *0.005= 17.672 ​$

Therefore, the population variance is computed as follows:

$σ ^ 2 ​ = ​ E(X ^2 )−E(X) ^ 2$

= $17.672−3.956 ^ 2$ = 2.0221

And finally, taking square root to the variance we get that the population standard deviation is $\sigma = \sqrt{ \sigma^2} = \sqrt{2.0221} = 1.422$