Math Problem Statement
Solution
The problem involves calculating the mean, variance, and standard deviation of a discrete random variable from a given probability distribution.
The table provided shows the values of and their corresponding probabilities :
X & 2 & 5 & 6 & 13 & 18 \\ P(X) & 0.11 & 0.09 & 0.11 & 0.14 & 0.55 \\ \end{array}$$ ### Step 1: Calculate the mean ($$\mu$$) or expected value $$E(X)$$ The mean is the expected value, calculated as: $$\mu = E(X) = \sum X_i \cdot P(X_i)$$ ### Step 2: Calculate the variance ($$\sigma^2$$) Variance is calculated as: $$\sigma^2 = E(X^2) - (E(X))^2$$ where $$E(X^2) = \sum X_i^2 \cdot P(X_i)$$ ### Step 3: Calculate the standard deviation ($$\sigma$$) The standard deviation is the square root of the variance: $$\sigma = \sqrt{\sigma^2}$$ I'll perform the calculations for you.Here are the calculated values based on the given dataHere are the calculated values based on the given - The mean (expected value) $$\mu$$ is approximately $$13.05$$. - The variance $$\sigma^2$$ is approximately $$38.2075$$. - The standard deviation $$\sigma$$ is approximately $$6.1812$$. Would you like more details or have any questions? Here are some related questions to further expand on the topic: 1. How is the expected value of a discrete random variable different from a continuous one? 2. Can variance ever be negative? Why or why not? 3. What is the significance of standard deviation in a probability distribution? 4. How does changing the probability distribution affect the variance? 5. How would you calculate the mean and variance if the values of $$X$$ were weighted differently? **Tip:** Remember, the expected value is a measure of central tendency, while variance and standard deviation give insights into how spread out the values are from the mean.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Statistics
Discrete Random Variables
Formulas
Mean (μ) = Σ[X * P(X)]
Variance (σ²) = Σ[(X² * P(X))] - μ²
Standard Deviation (σ) = √σ²
Theorems
Expected Value Theorem
Variance Formula
Suitable Grade Level
Grades 10-12
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