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?

---

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.