Math Problem Statement
Solve
What is the smallest value of x for which P(X ≤ x) > 0.5?
Solution
We are given the following PMF (Probability Mass Function) for the discrete random variable :
x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline p(x) & k & 3k & 5k & 7k & 9k & 11k & 13k & 15k & 17k \\ \end{array}$$ We are tasked with finding the smallest value of $$x$$ for which $$P(X \leq x) > 0.5$$. ### Step 1: Determine $$k$$ To find $$k$$, we use the fact that the sum of the probabilities must equal 1, i.e., $$p(0) + p(1) + p(2) + \dots + p(8) = 1$$ Substituting the values for $$p(x)$$: $$k + 3k + 5k + 7k + 9k + 11k + 13k + 15k + 17k = 1$$ Simplifying: $$(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17)k = 1$$ $$81k = 1$$ $$k = \frac{1}{81}$$ ### Step 2: Cumulative Probability $$P(X \leq x)$$ Now we calculate the cumulative probability $$P(X \leq x)$$ for each value of $$x$$ until it exceeds 0.5. - $$P(X \leq 0) = p(0) = k = \frac{1}{81}$$ - $$P(X \leq 1) = p(0) + p(1) = k + 3k = 4k = \frac{4}{81}$$ - $$P(X \leq 2) = k + 3k + 5k = 9k = \frac{9}{81} = \frac{1}{9}$$ - $$P(X \leq 3) = 9k + 7k = 16k = \frac{16}{81}$$ - $$P(X \leq 4) = 16k + 9k = 25k = \frac{25}{81}$$ - $$P(X \leq 5) = 25k + 11k = 36k = \frac{36}{81} = \frac{4}{9}$$ - $$P(X \leq 6) = 36k + 13k = 49k = \frac{49}{81}$$ - $$P(X \leq 7) = 49k + 15k = 64k = \frac{64}{81}$$ At $$P(X \leq 7) = \frac{64}{81}$$, this exceeds 0.5. ### Conclusion: The smallest value of $$x$$ for which $$P(X \leq x) > 0.5$$ is **7**. --- Would you like further details or have any additional questions? Here are 5 related questions for practice: 1. How would you interpret the cumulative distribution function (CDF) for this random variable? 2. How would the result change if the PMF values were uniformly distributed? 3. What is the expected value $$E(X)$$ of this random variable? 4. How do you calculate the variance for a discrete random variable given its PMF? 5. How can you compute the median for this discrete random variable? **Tip:** Always verify that the total probability adds up to 1 when working with probability distributions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Discrete Random Variables
Cumulative Probability
Probability Mass Function (PMF)
Formulas
Sum of probabilities: ∑ p(x) = 1
Cumulative probability: P(X ≤ x)
Theorems
Probability Theorem
Suitable Grade Level
Grades 10-12