The probability distribution of the random variable X represents the number of h

Step 1
a) The available information is shown in below:
$\begin{array}{|cc|}\hline x& P(x)\\ 0& 0.167\\ 1& 0.3289\\ 2& 0.2801\\ 3& 0.149\\ 4& 0.0382\\ 5& 0.0368\\ \hline\end{array}$
Calculate the theoretical mean of the random variable ${\displaystyle X}$
${\displaystyle Mean\left(\mu \right)=\sum x_{i}p\left(x_i\right)}$
${\displaystyle =0\left(0.167\right)+1\left(0.3289\right)+2\left(0.2801\right)+3\left(0.149\right)+4\left(0.0382\right)+5\left(0.0368\right)}$
${\displaystyle =0+0.3289+0.5602+0.447+0.1528+0.184}$
${\displaystyle =1.6729}$
The theoretical mean of the random variable X is 1.6729.
The theoretical mean of the random variable X for the given probability distribution is 1.6729.
Step 2
b) Calculate the theoretical standard deviation of the random variable ${\displaystyle X}$.
Standard deviation ${\displaystyle \left(\sigma \right)=\sqrt{Variance}}$
${\displaystyle =\sqrt{[\sum x_i^{2}p\left(x_i\right)]-{\left(\mu \right)}^2}}$
$=\sqrt{\left[\begin{array}{c}{0}^2(0.167)+1^2(0.3289)+2^2(0.2801)\\ +3^2(0.149)+4^2(0.0382)+5^2(0.0368)\end{array}\right]-(1.6729{)}^2}$
${\displaystyle =\sqrt{[0+0.3289+1.1204+1.341+0.6112+0.92]-\left(2.798594\right)}}$
${\displaystyle =\sqrt{4.3215-2.798594}}$
${\displaystyle =\sqrt{1.522906}}$
${\displaystyle =1.234061}$
The theoretical standard deviation of the random variable X is 1.234061
The theoretical standard deviation of the random variable X for the given probability distribution is 1.234061.
Step 3
c) The generated data is shown in below,
$\begin{array}{|ccccccccccccc|}\hline 1& 2& 4& 1& 0& 4& 1& 4& 0& 0& 2& 3& 2\\ 2& 2& 0& 3& 5& 0& 3& 2& 0& 2& 2& 1& \\ \hline\end{array}$
Calculate the approximate mean of the random variable ${\displaystyle X}$
${\displaystyle Mean\left(\bar{x}\right)=\frac{\sum x_i}{n}}$
${\displaystyle =\frac{1+2+4+\cdots +2+2+1}{25}}$
${\displaystyle =\frac{46}{25}}$
${\displaystyle =1.84}$
The approximate mean of the random variable X is 1.84.
The approximate mean of the random variable X for the given probability distribution is 1.84.
Step 4
d) Calculate the standard deviation of the random variable ${\displaystyle X}$
${\displaystyle s=\sqrt{Variance}}$
${\displaystyle =\sqrt{\frac{\sum {(x_i-\bar{x})}^2}{n-1}}}$