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!