Math Problem Statement

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

Solution

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

Step 1: Calculate the Expected Value (Mean) μ\mu

The expected value (mean) is given by:

μ=xiP(xi)\mu = \sum x_i P(x_i)

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

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

Step 2: Calculate the Variance

The variance σ2\sigma^2 is calculated by:

σ2=P(xi)(xiμ)2\sigma^2 = \sum P(x_i) \cdot (x_i - \mu)^2

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

σ2=(0.325×(01.225)2)+(0.388×(11.225)2)+(0.121×(21.225)2)+(0.069×(31.225)2)+(0.097×(41.225)2)\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 σ\sigma is the square root of the variance:

σ=σ2\sigma = \sqrt{\sigma^2}

I'll compute the full solution.### Results:

  1. Mean μ\mu: 1.225
  2. Variance σ2\sigma^2: 1.5444 (rounded to 4 decimal places)
  3. Standard Deviation σ\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.

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Statistics
Expected Value
Variance
Standard Deviation

Formulas

Expected Value: μ = Σ(x_i * P(x_i))
Variance: σ² = Σ(P(x_i) * (x_i - μ)²)
Standard Deviation: σ = √(σ²)

Theorems

Law of Total Expectation

Suitable Grade Level

Grades 10-12