To determine whether the given table can represent a probability distribution for the random variable $X$, the following two conditions must be satisfied:

1. All probabilities must be non-negative: Each value of $P(x)$ must satisfy $0 \leq P(x) \leq 1$.
2. The sum of the probabilities must equal 1: That is, $\sum P(x) = 1$.

Let's check these conditions.

1. Non-negativity check:

• $P(1) = 0.2639$
• $P(2) = 0.2605$
• $P(3) = 0.1585$
• $P(4) = 0.0985$
• $P(5) = 0.2186$

Since all of these values are between 0 and 1, the first condition is satisfied.

2. Sum of probabilities:

Let's add the given probabilities to check if they sum to 1: $0.2639 + 0.2605 + 0.1585 + 0.0985 + 0.2186 = 0.999999$

The sum is approximately 1, so the second condition is satisfied as well.

Conclusion:

Yes, the table can represent a probability distribution for the random variable $X$.

Would you like more details on probability distributions or have any other questions?

Here are 5 related questions to expand on this topic:

1. How can we calculate the mean of a probability distribution?
2. What is the variance of a probability distribution and how is it computed?
3. What is the difference between a probability mass function and a probability density function?
4. How do cumulative distribution functions relate to probability distributions?
5. What is the expected value and how does it relate to real-world scenarios?

Tip: Always check both the non-negativity and the sum of probabilities when verifying a probability distribution.