Question

Compute the mean and standard deviation of the random variable with the given discrete probability distribution.
x| -4 -1 6 8 9
P(x)| 0.24 0.14 0.27 0.26 0.09

Find the mean. Round the answer to 3 decimal places, if necessary.
Find the standard deviation. Round the answer to 3 decimal places, if necessary.

Answer 1

To compute the mean and standard deviation of a discrete probability distribution, we can use the following formulas:

1. **Mean (Expected Value)**:
\[
\mu = E(X) = \sum (x \cdot P(x))
\]

2. **Standard Deviation**:
First, we need to calculate the variance using the formula:
\[
\sigma^2 = Var(X) = \sum ((x - \mu)^2 \cdot P(x))
\]
Then, the standard deviation is the square root of the variance:
\[
\sigma = \sqrt{\sigma^2}
\]

### Step 1: Calculate the Mean

Using the given values for \(x\) and \(P(x)\):

\[
\begin{align*}
\mu & = (-4 \cdot 0.24) + (-1 \cdot 0.14) + (6 \cdot 0.27) + (8 \cdot 0.26) + (9 \cdot 0.09) \\
& = (-0.96) + (-0.14) + (1.62) + (2.08) + (0.81) \\
& = -0.96 - 0.14 + 1.62 + 2.08 + 0.81 \\
& = 3.41
\end{align*}
\]

### Mean:
\[
\mu = 3.41
\]

### Step 2: Calculate the Variance

Now we will need to calculate the variance using the mean we found (3.41):

\[
\begin{align*}
\sigma^2 & = ((-4 - 3.41)^2 \cdot 0.24) + ((-1 - 3.41)^2 \cdot 0.14) + ((6 - 3.41)^2 \cdot 0.27) + ((8 - 3.41)^2 \cdot 0.26) + ((9 - 3.41)^2 \cdot 0.09) \\
& = ((-7.41)^2 \cdot 0.24) + ((-4.41)^2 \cdot 0.14) + ((2.59)^2 \cdot 0.27) + ((4.59)^2 \cdot 0.26) + ((5.59)^2 \cdot 0.09) \\
& = (54.9281 \cdot 0.24) + (19.4481 \cdot 0.14) + (6.7081 \cdot 0.27) + (20.3681 \cdot 0.26) + (31.2481 \cdot 0.09) \\
& = 13.1887 + 2.7327 + 1.8092 + 5.2957 + 2.8111 \\
& = 25.0374
\end{align*}
\]

### Step 3: Calculate the Standard Deviation

Now we can compute the standard deviation:

\[
\sigma = \sqrt{\sigma^2} = \sqrt{25.0374} \approx 5.0037
\]

### Final Answers:

- Mean: \(\mu = 3.410\)
- Standard Deviation: \(\sigma \approx 5.004\)

Thus, after rounding to three decimal places, the answers are:

- Mean: **3.410**
- Standard Deviation: **5.004**