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.

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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.

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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.

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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.

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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!