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The image contains a table of values for a discrete random variable \( X \) with the following possible values for \( X \): \( -6, -2, 1, 4, 5 \). The task is to fill in the missing probability values \( P(X = x) \) 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: \( 0 \leq P(X = x) \leq 1 \).
2. The sum of the probabilities must equal 1: \( \sum P(X = x) = 1 \).

The provided table is:

\[
\begin{array}{|c|c|}
\hline
\text{Value of } X & P(X = x) \\
\hline
-6 & ? \\
-2 & ? \\
1 & 0.23 \\
4 & 0.30 \\
5 & 0.18 \\
\hline
\end{array}
\]

### 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.

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.

Learn how to fill in the missing probability values for the discrete random variable X with values -6, -2, 1, 4, and 5. This problem covers core probability distribution concepts and ensures that the sum of probabilities equals 1. Suitable for students in Grades 10-12.

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