Fill in the  values to give a legitimate probability distribution for the discrete random variable , whose possible values are 0,2,4,5, and 6.
Value x of X P(X=x)    0=0.23 4=0.13 6=0.23

To complete the probability distribution for the given discrete random variable \( X \), we need to assign probabilities \( P(X=x) \) to the values 2 and 5 such that the sum of all probabilities equals 1.

### Given:
- \( P(X=0) = 0.23 \)
- \( P(X=4) = 0.13 \)
- \( P(X=6) = 0.23 \)

### Let:
- \( P(X=2) = p \)
- \( P(X=5) = q \)

The sum of all probabilities for a legitimate probability distribution is:

\[
P(X=0) + P(X=2) + P(X=4) + P(X=5) + P(X=6) = 1
\]

Substitute the given values:

\[
0.23 + p + 0.13 + q + 0.23 = 1
\]

Simplify:

\[
0.59 + p + q = 1
\]

\[
p + q = 1 - 0.59
\]

\[
p + q = 0.41
\]

So, we need to find values for \( p \) and \( q \) such that their sum is 0.41. There are multiple pairs that satisfy this condition, but the simplest is to equally distribute the remaining probability:

\[
p = q = \frac{0.41}{2} = 0.205
\]

### Final Probability Distribution:
- \( P(X=0) = 0.23 \)
- \( P(X=2) = 0.205 \)
- \( P(X=4) = 0.13 \)
- \( P(X=5) = 0.205 \)
- \( P(X=6) = 0.23 \)

### Verification:
\[
0.23 + 0.205 + 0.13 + 0.205 + 0.23 = 1
\]

This confirms that the distribution is valid.

Would you like a detailed explanation of how to determine probabilities, or have any other questions? Here are some related questions:

1. How do you find the expected value of a discrete random variable?
2. What is the variance and standard deviation of a discrete random variable?
3. Can you explain the concept of cumulative distribution function (CDF)?
4. How do you check if a given set of probabilities is a legitimate distribution?
5. What is the difference between discrete and continuous probability distributions?

**Tip:** Always ensure that the sum of all probabilities in a distribution is equal to 1.

Fill in the values to give a legitimate probability distribution for the discrete random variable, whose possible values are 0, 2, 4, 5, and 6. Value x of X P(X=x) 0=0.23 4=0.13 6=0.23.

Learn how to complete the probability distribution for the discrete random variable with values 0, 2, 4, 5, and 6. This step-by-step guide demonstrates how to calculate the missing probabilities and verify the distribution. Suitable for students in Grades 9-12.

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