https://file.aiquickdraw.com/mathbot/file/1723446516167mxw1iuy2.jpg

Fill in the P(X=x) values to give a legitimate probability distribution for the discrete random variable X , whose possible values are 0,3,4,5, and 6.

To complete the table and make it represent a legitimate probability distribution, the sum of all probabilities \( P(X = x) \) must equal 1.

Given:
- \( P(X = 0) = 0.23 \)
- \( P(X = 5) = 0.23 \)
- \( P(X = 6) = 0.27 \)

Let the missing values be \( P(X = 3) = p_1 \) and \( P(X = 4) = p_2 \).

The total probability should sum to 1:

\[
0.23 + p_1 + p_2 + 0.23 + 0.27 = 1
\]

Simplifying, we get:

\[
p_1 + p_2 + 0.73 = 1
\]

\[
p_1 + p_2 = 0.27
\]

Now, the values of \( p_1 \) and \( p_2 \) must be non-negative and add up to 0.27. A simple solution could be:

\[
p_1 = 0.13 \quad \text{and} \quad p_2 = 0.14
\]

So, the completed table is:

\[
\begin{array}{|c|c|}
\hline
\text{Value } x \text{ of } X & P(X = x) \\
\hline
0 & 0.23 \\
3 & 0.13 \\
4 & 0.14 \\
5 & 0.23 \\
6 & 0.27 \\
\hline
\end{array}
\]

Would you like more details on the calculations or have any other questions?

---

Here are some related questions:

1. How do you calculate the mean (expected value) for this discrete probability distribution?
2. What is the cumulative distribution function (CDF) for the variable \( X \) in this distribution?
3. Can you explain how to calculate the variance of \( X \) for this distribution?
4. What would change in the distribution if \( P(X = 6) \) were instead 0.20?
5. How can this distribution be interpreted in a real-world scenario?
6. What other checks can be done to verify if a distribution is valid?
7. If you wanted to add a new possible value \( X = 2 \) to the distribution, how would that affect the probabilities?
8. What are the implications of having two probabilities be exactly equal in a distribution?

**Tip:** When distributing probabilities, ensure that the sum of the assigned probabilities equals 1 and that each individual probability is non-negative and does not exceed 1.

Learn how to complete a legitimate probability distribution for a discrete random variable X with specific values. Understand the concept of probability distributions and how to calculate missing probabilities. Suitable for undergraduate students studying probability theory and discrete random variables.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation