Question

use the probability distribution for the random variable x to answer the question.

\\(\

$$\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \\\\ \\hline p(x) & 0.12 & 0.35 & 0.35 & 0.06 & 0.12 \\end{array}$$

\\)

calculate the population mean, variance, and standard deviation. (round your standard deviation to three decimal places.)

mean
variance
standard deviation

Explanation:

Calculate the population mean

Using the Mean Calculation and Discrete Random Variables knowledge points

$$
LATEXBLOCK0
$$

Calculate the expected value of x squared

To find the variance, we first compute \(E(x^2)\):

$$
LATEXBLOCK1
$$

Calculate the population variance

The variance \(\sigma^2\) is calculated using the formula \(\sigma^2 = E(x^2) - \mu^2\):

$$
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$$

Rounding to three decimal places (or keeping it exact if needed, since \(1.2859\) is exact):

$$ \sigma^2 \approx 1.286 $$

Calculate the population standard deviation

The standard deviation \(\sigma\) is the square root of the variance:

$$
LATEXBLOCK3
$$

Answer:

• mean: \(1.71\)
• variance: \(1.2859\) (or \(1.286\))
• standard deviation: \(1.134\)