Fill in P= (X=x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are 1, 3, 4,5, and 6.    Value x of X P(X=X) 

To create a legitimate probability distribution for a discrete random variable \( X \) that takes values \( 1, 3, 4, 5, 6 \), we need to assign probabilities \( P(X = x) \) to each value of \( x \) such that the following conditions hold:

1. The probability of each outcome must be between 0 and 1: 
   \[
   0 \leq P(X = x) \leq 1
   \]
2. The sum of the probabilities for all possible outcomes must equal 1:
   \[
   P(X = 1) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) = 1
   \]

### Let's fill in some sample probabilities that meet these conditions:

| Value \( x \) of \( X \) | \( P(X = x) \)  |
|-------------------------|----------------|
| 1                       | 0.2            |
| 3                       | 0.1            |
| 4                       | 0.3            |
| 5                       | 0.25           |
| 6                       | 0.15           |

### Verification:
\[
0.2 + 0.1 + 0.3 + 0.25 + 0.15 = 1
\]

This satisfies both conditions, so this is a legitimate probability distribution for the discrete random variable \( X \).

Would you like more details or have any other questions?

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Here are 5 related questions to expand on this:
1. How would the distribution change if \( P(X = 1) \) were increased?
2. Can you explain why the sum of probabilities must always be 1?
3. How do we interpret the probability \( P(X = 4) = 0.3 \)?
4. What is the expectation (mean) of \( X \) for this probability distribution?
5. How would you calculate the variance of \( X \)?

**Tip:** The probabilities must sum to 1 because the total probability space accounts for all possible outcomes of the random variable.

Fill in P= (X=x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are 1, 3, 4, 5, and 6. Value x of X P(X=X)

Learn how to assign probabilities to the values 1, 3, 4, 5, and 6 for a discrete random variable X to create a legitimate probability distribution. This solution explains the conditions, including the sum of probabilities equaling 1, and provides a sample distribution with verification. Ideal for students in Grades 9-12 studying probability.

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