exercise 2.the number of quarantine protocol violators in one district of a city is shown in the table. find the variance and standard deviation.

Question

Accepted Answer

# Explanation:
## Step1: Calculate the expected value $E(X)$
The formula for the expected value of a discrete random variable is $E(X)=\sum_{i}x_{i}P(x_{i})$.
$$
\begin{align*}
E(X)&=(25\times\frac{3}{20})+(30\times\frac{9}{20})+(35\times\frac{7}{20})+(40\times\frac{1}{20})\
&=\frac{75 + 270+245 + 40}{20}\
&=\frac{630}{20}=31.5
\end{align*}
$$
## Step2: Calculate $E(X^{2})$
$$
\begin{align*}
E(X^{2})&=(25^{2}\times\frac{3}{20})+(30^{2}\times\frac{9}{20})+(35^{2}\times\frac{7}{20})+(40^{2}\times\frac{1}{20})\
&=(625\times\frac{3}{20})+(900\times\frac{9}{20})+(1225\times\frac{7}{20})+(1600\times\frac{1}{20})\
&=\frac{1875+8100 + 8575+1600}{20}\
&=\frac{20150}{20}=1007.5
\end{align*}
$$
## Step3: Calculate the variance $\text{Var}(X)$
The formula for variance is $\text{Var}(X)=E(X^{2})-[E(X)]^{2}$
$$
\begin{align*}
\text{Var}(X)&=1007.5-(31.5)^{2}\
&=1007.5 - 992.25\
&=15.25
\end{align*}
$$
## Step4: Calculate the standard deviation $\sigma$
The formula for standard deviation is $\sigma=\sqrt{\text{Var}(X)}$
$$
\sigma=\sqrt{15.25}\approx3.905
$$

# Answer:
The variance is $15.25$ and the standard deviation is approximately $3.905$

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Question

Explanation:

Step1: Calculate the expected value \(E(X)\)

Step2: Calculate \(E(X^{2})\)

Step3: Calculate the variance \(\text{Var}(X)\)

Step4: Calculate the standard deviation \(\sigma\)

Answer: