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 with its corresponding probabilities , we will use the following steps:
Step 1: Calculate the Expected Value (Mean)
The expected value (mean) is given by:
Where are the values of the random variable and are the corresponding probabilities.
Step 2: Calculate the Variance
The variance is calculated by:
Substitute the values of , , and :
Let's compute this step-by-step.
Step 3: Calculate the Standard Deviation
Finally, the standard deviation is the square root of the variance:
I'll compute the full solution.### Results:
- Mean : 1.225
- Variance : 1.5444 (rounded to 4 decimal places)
- Standard Deviation : 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:
- How do you interpret the standard deviation in a probability distribution?
- How would the variance change if one of the probabilities were higher or lower?
- What is the significance of a small vs large standard deviation?
- How does the standard deviation compare to the mean for different distributions?
- 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
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