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Problem 10

# Calculate the standard deviation of $$X$$ for each probability distribution. (You calculated the expected values in the last exercise set. Round all answers to two decimal places.) $$\begin{array}{|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & .1 & .2 & .5 & .2 \\ \hline \end{array}$$

Expert verified
The standard deviation of the given probability distribution is approximately $$1.14$$.
See the step by step solution

## Step 1: Review probability distribution table

The given probability distribution table looks as follows: $$\begin{array}{|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 \\\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & .1 & .2 & .5 & .2 \\ \hline \end{array}$$

## Step 2: Calculate E(X) from previous exercise

From the previous exercise, we've calculated the expected value E(X) as: E(X) = 1 * 0.1 + 2 * 0.2 + 3 * 0.5 + 4 * 0.2 = 2.7

## Step 3: Calculate E(X^2)

Now, we need to calculate E(X^2) using the table of probabilities: E(X^2) = $$1^2 * 0.1 + 2^2 * 0.2 + 3^2 * 0.5 + 4^2 * 0.2$$ = 0.1 + 0.8 + 4.5 + 3.2 = 8.6

## Step 4: Calculate Variance

Variance is calculated as Variance = E(X^2) - (E(X))^2: Variance = 8.6 - (2.7)^2 = 8.6 - 7.29 = 1.31

## Step 5: Calculate Standard Deviation

Standard deviation is the square root of the variance: Standard Deviation = $$\sqrt{1.31}$$ ≈ 1.14 The standard deviation of the given probability distribution is approximately 1.14.

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