Question A survey asked a random sample of students taking complex algebra how many questions were marked incorrect on their last quiz. The table below represents the probability density function for the random variable $X$, the number of questions marked incorrect. Find the standard deviation of $X$. - Round the final answer to two decimal places. \begin{tabular}{|c|c|} \hline$x$ & $P(X=x)$ \\ \hline 1 & $1 / 4$ \\ \hline 3 & $1 / 4$ \\ \hline 4 & $1 / 4$ \\ \hline \end{tabular}
Solution
Step 1 ：First, we need to calculate the expected value (mean) of $X$, which is given by $E[X] = \sum xP(X=x)$, where the sum is over all possible values of $X$. In this case, the possible values of $X$ are 1, 3, and 4, and the corresponding probabilities are all $1 / 4$. So, $E[X] = 1*(1/4) + 3*(1/4) + 4*(1/4) = 2.0$.
Step 2 ：Next, we calculate the expected value of the squared deviation of $X$ from its mean, which is given by $E[(X - E[X])^2] = \sum (x - E[X])^2P(X=x)$, where the sum is over all possible values of $X$. So, $E[(X - E[X])^2] = (1 - 2)^2*(1/4) + (3 - 2)^2*(1/4) + (4 - 2)^2*(1/4) = 1.5$.
Step 3 ：Finally, we take the square root of the variance to get the standard deviation. So, the standard deviation of $X$ is $\sqrt{1.5} = 1.224744871391589$.
Step 4 ：Rounding the final answer to two decimal places, we get \(\boxed{1.22}\).
