
 
 
 
 
 
 
     *x*

15     

22     

34     

40     

*P*(*X* = *x*)

0.18     

0.30     

0.29     

0.23     
 
 
**a.**

Is this a valid probability distribution?
 
 
 
multiple choice

Yes

No
 
 
**Round answers to two decimal places.**
 
**b.**

What is the probability that the random variable *X* is less than 35? 
 
  Probability
  
 
**c.**

What is the probability that the random variable *X* is between 13 and 28? 
 
  Probability
  
 
**d.**

What is the probability that the random variable *X* is greater than 21? 
 
  Probability
  

### Part (a): Is this a valid probability distribution?

A probability distribution is valid if the sum of the probabilities equals 1 and all probabilities are between 0 and 1.

Given probabilities:

\[
P(X = 15) = 0.18, \quad P(X = 22) = 0.30, \quad P(X = 34) = 0.29, \quad P(X = 40) = 0.23
\]

Sum of the probabilities:

\[
0.18 + 0.30 + 0.29 + 0.23 = 1.00
\]

Since the sum is exactly 1 and all probabilities are between 0 and 1, **this is a valid probability distribution**.

The correct answer to part (a) is **Yes**.

---

### Part (b): Probability that \( X \) is less than 35

We are looking for \( P(X < 35) \), which means summing the probabilities for \( X = 15 \) and \( X = 22 \):

\[
P(X < 35) = P(X = 15) + P(X = 22) = 0.18 + 0.30 = 0.48
\]

The probability that \( X \) is less than 35 is **0.48**.

---

### Part (c): Probability that \( X \) is between 13 and 28

We are looking for \( P(13 < X < 28) \), which is the probability for \( X = 15 \) and \( X = 22 \):

\[
P(13 < X < 28) = P(X = 15) + P(X = 22) = 0.18 + 0.30 = 0.48
\]

The probability that \( X \) is between 13 and 28 is **0.48**.

---

### Part (d): Probability that \( X \) is greater than 21

We are looking for \( P(X > 21) \), which means summing the probabilities for \( X = 22 \), \( X = 34 \), and \( X = 40 \):

\[
P(X > 21) = P(X = 22) + P(X = 34) + P(X = 40) = 0.30 + 0.29 + 0.23 = 0.82
\]

The probability that \( X \) is greater than 21 is **0.82**.

---

Do you want more details on any part of this? Here are some follow-up questions:

1. What properties define a valid probability distribution?
2. How would the solution change if the sum of probabilities exceeded 1?
3. How would you calculate probabilities for a continuous random variable?
4. What other types of probability distributions are there besides discrete?
5. How do we calculate probabilities when the distribution isn't uniform?

**Tip:** Always check that the sum of probabilities equals 1 when working with discrete probability distributions!

Is this a valid probability distribution? What is the probability that the random variable X is less than 35? What is the probability that the random variable X is between 13 and 28? What is the probability that the random variable X is greater than 21?

Explore how to validate a discrete probability distribution by checking if the sum of probabilities equals 1. Learn how to calculate the probability that a random variable X is less than 35, between 13 and 28, and greater than 21. This problem involves basic probability concepts and formulas, suitable for students in Grades 9-12.

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