Math Problem Statement
The following table provides a probability distribution for the random variable x. Compute E(x), the expected value of x. Compute σ^2, the variance of x (to 1 decimal). Compute σ, the standard deviation of x (to 2 decimals).
Solution
The problem provides a probability distribution for the random variable with the following values:
x & f(x) \\ \hline 3 & 0.25 \\ 6 & 0.50 \\ 9 & 0.25 \\ \end{array}$$ We need to calculate: 1. $$E(x)$$ - The expected value of $$x$$ 2. $$\sigma^2$$ - The variance of $$x$$ (to 1 decimal place) 3. $$\sigma$$ - The standard deviation of $$x$$ (to 2 decimal places) --- ### Step (a): Expected Value $$E(x)$$ The expected value $$E(x)$$ is given by the formula: $$E(x) = \sum (x_i \cdot f(x_i))$$ Using the provided values of $$x$$ and $$f(x)$$: $$E(x) = (3 \cdot 0.25) + (6 \cdot 0.50) + (9 \cdot 0.25)$$ $$E(x) = 0.75 + 3.00 + 2.25 = 6.00$$ Thus, the expected value $$E(x) = 6$$. --- ### Step (b): Variance $$\sigma^2$$ The variance $$\sigma^2$$ is calculated as: $$\sigma^2 = \sum \left[ (x_i - E(x))^2 \cdot f(x_i) \right]$$ Substituting $$E(x) = 6$$: $$\sigma^2 = (3 - 6)^2 \cdot 0.25 + (6 - 6)^2 \cdot 0.50 + (9 - 6)^2 \cdot 0.25$$ $$\sigma^2 = (-3)^2 \cdot 0.25 + (0)^2 \cdot 0.50 + (3)^2 \cdot 0.25$$ $$\sigma^2 = 9 \cdot 0.25 + 0 \cdot 0.50 + 9 \cdot 0.25 = 2.25 + 0 + 2.25 = 4.5$$ Thus, the variance $$\sigma^2 = 4.5$$. --- ### Step (c): Standard Deviation $$\sigma$$ The standard deviation $$\sigma$$ is the square root of the variance: $$\sigma = \sqrt{\sigma^2} = \sqrt{4.5} \approx 2.12$$ Thus, the standard deviation $$\sigma \approx 2.12$$ (rounded to 2 decimal places). --- Would you like more details on any of these steps or have any further questions? Here are 5 related questions to explore: 1. How does changing the values of $$f(x)$$ affect the expected value? 2. Can the expected value be a decimal even if $$x$$ values are integers? 3. What happens to the variance if all $$f(x)$$ values are the same? 4. Why is the standard deviation always a non-negative number? 5. How do the variance and standard deviation relate to the spread of a distribution? **Tip:** The standard deviation is a more intuitive measure of spread because it has the same units as the original data.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Expected Value
Variance
Standard Deviation
Formulas
E(x) = Σ(x_i * f(x_i))
σ^2 = Σ[(x_i - E(x))^2 * f(x_i)]
σ = √σ^2
Theorems
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Suitable Grade Level
Grades 11-12
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