Answer
Given below.
Work Step by Step
(a)
The discrete probability distribution is obtained as
\[x\] Frequency Probability
0 10 0.0111
1 30 0.0333
2 520 0.5778
3 250 0.2778
4 70 0.0778
5 17 0.0189
6 3 0.0033
The probability of each value of the variable can be calculated by using the formula:
\[P\left( {{x}_{i}} \right)=\frac{{{f}_{i}}}{N}\]
Here, \[{{f}_{i}}\]is the frequency of each value, \[N\]is the total frequency, and \[P\left( {{x}_{i}} \right)\]is the probability of each value of the random variable where \[{{x}_{i}}\] is the value of the random variable.
The total frequency is calculated as
\[\begin{align}
& N=\sum{{{f}_{i}}} \\
& =10+30+\cdots +3 \\
& =900
\end{align}\]
(b)
The graph is shown below:
The below graph shows the probability of each value of the random variable. The horizontal axis shows the number of ideal children and the vertical axis shows the probability. The origin is at the point\[\left( 0,0 \right)\]. The probability is plotted corresponding to each value and a perpendicular line is drawn from the point to the horizontal axis.
(c)
The number of people waiting in a queue during lunchtime is the random variable in the provided probability distribution, which is represented by \[X\]. The mean of the random variable \[X\]is calculated by adding the multiplied values of each random variable and their corresponding probability.
The required mean is calculated as
\[\begin{align}
& {{\mu }_{X}}=\sum{\left( x\cdot P\left( x \right) \right)} \\
& =\left( 0\times 0.0111 \right)+\left( 1\times 0.0333 \right)+\cdots +\left( 6\times 0.0033 \right) \\
& =2.448
\end{align}\]
Therefore, the mean of the random variable \[X\]is 2.448 people.
To record the value of the random variable, an experiment is repeated many times. The average of the outcomes of the experiment is considered as the mean or expected value of the discrete random variable. If the data are collected from many families, then the expected number of ideal children in each family is 2.448 children.
(d)
The standard deviation of the random variable \[X\]can be calculated using the formula:
\[\begin{align}
& {{\sigma }_{X}}=\sqrt{\sum{\left( {{\left( x-{{\mu }_{X}} \right)}^{2}}\cdot P\left( x \right) \right)}} \\
& =\sqrt{\sum{\left( {{x}^{2}}\cdot P\left( x \right) \right)-\mu _{X}^{2}}}
\end{align}\]
Where\[{{\sigma }_{X}}\]is the standard deviation of\[X\], \[{{\mu }_{X}}\]is the mean of\[X\], and \[P\left( x \right)\]is the probability of each value of the random variable.
The required standard deviation can be calculated as
\[\begin{align}
& {{\sigma }_{X}}=\sqrt{\sum{\left( {{x}^{2}}\cdot P\left( x \right) \right)-\mu _{X}^{2}}} \\
& =\sqrt{\left( \left( {{0}^{2}}\times 0.0111 \right)+\left( {{1}^{2}}\times 0.0333 \right)+\cdots +\left( {{6}^{2}}\times 0.0033 \right) \right)-{{2.448}^{2}}} \\
& =0.8296
\end{align}\]
