Find the standard deviation of the random variable.
-The random variable X is the number of siblings of a student selected at random from a particular secondary school. Its probability distribution is given in the table. Round the answer to three Decimal places when necessary. $\begin{array}{r|cccccc}
\mathrm{x} & 0 & 1 & 2 & 3 & 4 & 5 \
\hline \mathrm{P}(\mathrm{X}=\mathrm{x}) & \frac{1}{3} & \frac{13}{48} & \frac{1}{6} & \frac{7}{48} & \frac{1}{24} & \frac{1}{24}
\end{array}$
A)$1.006$
B) $2.012$
C) $1.397$
D) $1.676$

Question

Quizplus · Accepted Answer

The standard deviation of a random variable X, denoted as $\sigma$, is calculated using the formula $\sigma = \sqrt{E(X^2) - [E(X)]^2}$, where $E(X)$ is the expected value of X and $E(X^2)$ is the expected value of $X^2$. First, calculate the expected value $E(X)$ and $E(X^2)$:$E(X) = (0)\left(\frac{1}{3}\right) + (1)\left(\frac{13}{48}\right) + (2)\left(\frac{1}{6}\right) + (3)\left(\frac{7}{48}\right) + (4)\left(\frac{1}{24}\right) + (5)\left(\frac{1}{24}\right)$$E(X) = 0 + \frac{13}{48} + \frac{1}{3} + \frac{7}{16} + \frac{1}{6} + \frac{5}{24}$$E(X) = \frac{13}{48} + \frac{16}{48} + \frac{21}{48} + \frac{4}{48} + \frac{5}{24}$$E(X) = \frac{54}{48} + \frac{10}{48}$$E(X) = \frac{64}{48} = \frac{4}{3}$Then, calculate $E(X^2)$:$E(X^2) = (0)^2\left(\frac{1}{3}\right) + (1)^2\left(\frac{13}{48}\right) + (2)^2\left(\frac{1}{6}\right) + (3)^2\left(\frac{7}{48}\right) + (4)^2\left(\frac{1}{24}\right) + (5)^2\left(\frac{1}{24}\right)$$E(X^2) = 0 + \frac{13}{48} + \frac{4}{6} + \frac{63}{48} + \frac{16}{24} + \frac{25}{24}$$E(X^2) = \frac{13}{48} + \frac{32}{48} + \frac{63}{48} + \frac{41}{24}$$E(X^2) = \frac{108}{48} + \frac{41}{24}$$E(X^2) = \frac{108}{48} + \frac{82}{48}$$E(X^2) = \frac{190}{48} = \frac{95}{24}$Now, calculate the variance $\sigma^2 = E(X^2) - [E(X)]^2$:$\sigma^2 = \frac{95}{24} - \left(\frac{4}{3}\right)^2$$\sigma^2 = \frac{95}{24} - \frac{16}{9}$To find a common denominator, convert fractions:$\sigma^2 = \frac{95}{24} - \frac{32}{24}$$\sigma^2 = \frac{63}{24} = \frac{21}{8}$Finally, calculate the standard deviation $\sigma = \sqrt{\frac{21}{8}}$:$\sigma \approx 1.397$Therefore, the standard deviation of the random variable X is approximately 1.397.

Find the Standard Deviation of the Random Variable A) 1.0061.0061.006 B) 2.0122.0122.012

Find the Standard Deviation of the Random Variable A) $1.006$ B) $2.012$