Math Problem Statement

Fill in the missing P(X = x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are -6, -2, 1, 4, and 5.

Solution

The image contains a table of values for a discrete random variable XX with the following possible values for XX: −6,−2,1,4,5-6, -2, 1, 4, 5. The task is to fill in the missing probability values P(X=x)P(X = x) so that the given probability distribution becomes legitimate.

To ensure a legitimate probability distribution, the probabilities must satisfy two conditions:

  1. Each probability must be between 0 and 1: 0≤P(X=x)≤10 \leq P(X = x) \leq 1.
  2. The sum of the probabilities must equal 1: ∑P(X=x)=1\sum P(X = x) = 1.

The provided table is:

\hline \text{Value of } X & P(X = x) \\ \hline -6 & ? \\ -2 & ? \\ 1 & 0.23 \\ 4 & 0.30 \\ 5 & 0.18 \\ \hline \end{array}$$ ### Step 1: Calculate the sum of known probabilities. $$0.23 + 0.30 + 0.18 = 0.71$$ ### Step 2: Calculate the remaining probability for $$X = -6$$ and $$X = -2$$. The remaining probability to distribute between $$X = -6$$ and $$X = -2$$ is: $$1 - 0.71 = 0.29$$ Let the probabilities for $$X = -6$$ and $$X = -2$$ be $$p_1$$ and $$p_2$$, respectively. We know that: $$p_1 + p_2 = 0.29$$ ### Step 3: Assign reasonable values to $$p_1$$ and $$p_2$$. Without further constraints, we can assign equal or reasonable values that sum to 0.29. One common approach is to split them equally, so: $$p_1 = p_2 = 0.145$$ Thus, the completed table would be: $$\begin{array}{|c|c|} \hline \text{Value of } X & P(X = x) \\ \hline -6 & 0.145 \\ -2 & 0.145 \\ 1 & 0.23 \\ 4 & 0.30 \\ 5 & 0.18 \\ \hline \end{array}$$ This satisfies the condition that the probabilities sum to 1 and are all between 0 and 1. Would you like more details or have any questions? Here are 5 relative questions for further exploration: 1. How are probability distributions used to represent random variables? 2. What conditions must a discrete probability distribution always satisfy? 3. Can you provide an example of an invalid probability distribution? 4. How do you calculate expected value from a probability distribution? 5. What is the difference between a probability mass function and a probability density function? **Tip:** When creating a probability distribution, always ensure the probabilities sum to 1 to reflect the total certainty of all possible outcomes occurring.

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Math Problem Analysis

Mathematical Concepts

Probability Distributions
Discrete Random Variables

Formulas

P(X = x) values must sum to 1

Theorems

Basic Probability Theorem
Discrete Distribution Rules

Suitable Grade Level

Grades 10-12

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