Compute the standard deviation of the random variable x. x=0,1,2,3,4 P(x)=0.325,0.388,0.121,0.069,0.097

To compute the standard deviation of the random variable \( x \) with its corresponding probabilities \( P(x) \), we will use the following steps:

### Step 1: Calculate the Expected Value (Mean) \( \mu \)
The expected value (mean) is given by:

\[
\mu = \sum x_i P(x_i)
\]

Where \( x_i \) are the values of the random variable and \( P(x_i) \) are the corresponding probabilities.

\[
\mu = (0 \times 0.325) + (1 \times 0.388) + (2 \times 0.121) + (3 \times 0.069) + (4 \times 0.097)
\]
\[
\mu = 0 + 0.388 + 0.242 + 0.207 + 0.388 = 1.225
\]

### Step 2: Calculate the Variance
The variance \( \sigma^2 \) is calculated by:

\[
\sigma^2 = \sum P(x_i) \cdot (x_i - \mu)^2
\]

Substitute the values of \( x_i \), \( P(x_i) \), and \( \mu = 1.225 \):

\[
\sigma^2 = (0.325 \times (0 - 1.225)^2) + (0.388 \times (1 - 1.225)^2) + (0.121 \times (2 - 1.225)^2) + (0.069 \times (3 - 1.225)^2) + (0.097 \times (4 - 1.225)^2)
\]

Let's compute this step-by-step.

### Step 3: Calculate the Standard Deviation
Finally, the standard deviation \( \sigma \) is the square root of the variance:

\[
\sigma = \sqrt{\sigma^2}
\]

I'll compute the full solution.### Results:
1. **Mean \( \mu \)**: 1.225
2. **Variance \( \sigma^2 \)**: 1.5444 (rounded to 4 decimal places)
3. **Standard Deviation \( \sigma \)**: 1.243 (rounded to 3 decimal places)

Would you like more details on any part of this solution?

Here are 5 related questions to expand your understanding:
1. How do you interpret the standard deviation in a probability distribution?
2. How would the variance change if one of the probabilities were higher or lower?
3. What is the significance of a small vs large standard deviation?
4. How does the standard deviation compare to the mean for different distributions?
5. Can you calculate the coefficient of variation using this standard deviation?

**Tip**: Variance measures the spread of data points around the mean, while standard deviation is its square root, providing a more intuitive measure of dispersion.

Learn how to compute the standard deviation for the random variable x = 0,1,2,3,4 with probabilities P(x)=0.325,0.388,0.121,0.069,0.097. This step-by-step solution includes calculating the mean, variance, and standard deviation. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation