Fill in the P (X-x) values to give a legitimate probability, distribution for the discrete random variable X, whose possible values are 1, 2, 3, 5, and 6.

Value x of X

P(X= x)

-1

0.15

2

3

0.19

5

0.10

6

To create a legitimate probability distribution, the sum of all the probabilities \( P(X = x) \) must equal 1.

Given the probabilities for the values \( X = 1, 3, 5, 6 \) as 0.15, 0.19, and 0.10 respectively, let's determine the missing probabilities for \( X = 2 \) and \( X = 6 \).

### Step 1: Write down the given probabilities
\[
P(X = 1) = 0.15 \\
P(X = 3) = 0.19 \\
P(X = 5) = 0.10 \\
P(X = 6) = ?
\]

### Step 2: Let the missing probabilities be denoted as \( P(X = 2) \) and \( P(X = 6) \).
Let:
\[
P(X = 2) = p_2 \\
P(X = 6) = p_6
\]

### Step 3: Use the condition that the sum of probabilities must equal 1.
\[
P(X = 1) + p_2 + P(X = 3) + P(X = 5) + p_6 = 1
\]
Substitute the known values:
\[
0.15 + p_2 + 0.19 + 0.10 + p_6 = 1
\]
Simplify the equation:
\[
0.44 + p_2 + p_6 = 1
\]
\[
p_2 + p_6 = 1 - 0.44 = 0.56
\]

### Step 4: Determine the values for \( P(X = 2) \) and \( P(X = 6) \).
You have some freedom in choosing \( p_2 \) and \( p_6 \) as long as their sum is 0.56. Here’s one possible solution:
- Let \( P(X = 2) = 0.36 \)
- Let \( P(X = 6) = 0.20 \)

### Final Probability Distribution:
\[
\begin{array}{|c|c|}
\hline
X & P(X = x) \\
\hline
1 & 0.15 \\
2 & 0.36 \\
3 & 0.19 \\
5 & 0.10 \\
6 & 0.20 \\
\hline
\end{array}
\]
The sum of all probabilities now equals 1.

Let me know if you need further details or have any questions.

**Related Questions:**
1. What is a probability distribution?
2. How can you verify if a given set of probabilities is legitimate?
3. What is the significance of the sum of probabilities being equal to 1?
4. How would you calculate the expected value of \( X \) with this distribution?
5. How would you calculate the variance of \( X \) with this distribution?
6. What happens if the sum of the probabilities is not equal to 1?
7. What are some examples of discrete random variables in real life?
8. How does a probability mass function (PMF) relate to this problem?

**Tip:** When dealing with probability distributions, always verify that the total probability equals 1 to ensure the distribution is valid.

Learn how to create a legitimate probability distribution for a discrete random variable X with given probabilities for values 1, 3, 5, and 6. Understand the importance of the sum of probabilities being equal to 1 and solve for missing probabilities.

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