
 
 
 
 
 
 
     *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
  

Let's break down the problem and solve it step by step.

### Part (a) — Is this a valid probability distribution?
A valid probability distribution must satisfy two conditions:
1. **Each probability must be between 0 and 1**, inclusive.
2. **The sum of all probabilities must equal 1**.

Given the probabilities:

\[
P(X = x) = 0.18, 0.30, 0.29, 0.23
\]

Let's check the sum of these probabilities:

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

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

So, the answer to part (a) is:  
**Yes**.

---

### Part (b) — What is the probability that the random variable \( X \) is less than 35?
The values of \( X \) less than 35 are \( X = 15 \) and \( X = 22 \). We need to sum their probabilities:

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

So, the probability that \( X \) is less than 35 is:  
**0.48**.

---

### Part (c) — What is the probability that the random variable \( X \) is between 13 and 28?
The values of \( X \) between 13 and 28 are \( X = 15 \) and \( X = 22 \). We need to sum their probabilities:

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

So, the probability that \( X \) is between 13 and 28 is:  
**0.48**.

---

### Part (d) — What is the probability that the random variable \( X \) is greater than 21?
The values of \( X \) greater than 21 are \( X = 22 \), \( X = 34 \), and \( X = 40 \). We need to sum their probabilities:

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

So, the probability that \( X \) is greater than 21 is:  
**0.82**.

---

Let me know if you'd like more details or clarifications! Here are 5 related questions you might find interesting:

1. What conditions must a probability distribution always satisfy?
2. How would you calculate the expected value (mean) of this probability distribution?
3. What is the probability that \( X \) is exactly equal to 34?
4. How would the probabilities change if the sum was greater than 1?
5. How do you interpret the sum of probabilities in a discrete distribution?

**Tip:** Always check if the sum of probabilities equals 1 when dealing with distributions!

The query submitted by the user:

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

b. What is the probability that the random variable X is less than 35?

c. What is the probability that the random variable X is between 13 and 28?

d. What is the probability that the random variable X is greater than 21?

Learn how to determine if a probability distribution is valid, calculate the probability of a random variable X being less than 35, between 13 and 28, and greater than 21. This detailed solution covers essential probability concepts and is suitable for students in Grades 9-12.

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