Math Problem Statement

The discrete random variable X has the probability function defined as P(X = x) = kx for x = 2, 4, 6; P(X = x) = k(x - 2) for x = 8; and 0 otherwise. Show that k = 1/18.

Solution

To solve part (a) and demonstrate that k=118k = \frac{1}{18}, we use the fact that the total probability of a discrete random variable must sum to 1:

∑xP(X=x)=1.\sum_x P(X = x) = 1.

From the problem, the probability function P(X=x)P(X = x) is defined as:

\begin{cases} kx, & x = 2, 4, 6, \\ k(x-2), & x = 8, \\ 0, & \text{otherwise.} \end{cases}$$ ### Step-by-Step Solution 1. **Write the total probability equation**: Substitute the values of $$x$$ where $$P(X = x) \neq 0$$ (i.e., $$x = 2, 4, 6, 8$$): $$\sum_{x} P(X = x) = P(X = 2) + P(X = 4) + P(X = 6) + P(X = 8).$$ Using the function definition: $$P(X = 2) = k(2), \, P(X = 4) = k(4), \, P(X = 6) = k(6), \, P(X = 8) = k(8 - 2).$$ Thus: $$\sum_{x} P(X = x) = 2k + 4k + 6k + 6k.$$ 2. **Simplify the equation**: Combine terms: $$\sum_{x} P(X = x) = 18k.$$ 3. **Set the total probability to 1**: Since the total probability must equal 1: $$18k = 1.$$ 4. **Solve for $$k$$**: Divide both sides by 18: $$k = \frac{1}{18}.$$ ### Conclusion: This verifies that $$k = \frac{1}{18}$$. Would you like further explanation or assistance with related parts of this problem? Here are 5 follow-up questions to extend your understanding: 1. How do we determine the range of $$x$$ values for which $$P(X = x)$$ is non-zero? 2. Why does the total probability of all possible outcomes equal 1 in any probability distribution? 3. How would the value of $$k$$ change if additional $$x$$-values were added to the probability function? 4. What are the implications of a uniform versus non-uniform probability function in a random variable? 5. Can you identify real-world scenarios modeled by discrete random variables with piecewise probability functions? **Tip**: Always verify that the sum of probabilities in a discrete random variable adds up to 1—this ensures the function is a valid probability distribution.

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Math Problem Analysis

Mathematical Concepts

Probability
Discrete Random Variables
Normalization of Probability

Formulas

Sum of probabilities: Σ P(X = x) = 1
Piecewise probability function: P(X = x)

Theorems

Normalization Property of Probability Distributions

Suitable Grade Level

Grades 11-12 or Introductory College Level

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