The problem asks us to determine three things based on the probability density function (PDF) of a random variable $X$:

1. Find $k$
2. Find $P(X < 4)$
3. Find $P(3 < X \leq 6)$

The values of $X$ and the corresponding probabilities are given in the following form:

$X: 0, 1, 2, 3, 4, 5, 6$ $P(X = x): k, 3k, 5k, 7k, 9k, 11k, 13k$

Step 1: Find $k$

Since the total probability of a discrete random variable must sum to 1, we can set up the following equation:

$P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) = 1$

Substituting the probabilities in terms of $k$:

$k + 3k + 5k + 7k + 9k + 11k + 13k = 1$

Simplifying the left-hand side:

$(1 + 3 + 5 + 7 + 9 + 11 + 13)k = 1$

$49k = 1$

Solving for $k$:

$k = \frac{1}{49}$

Step 2: Find $P(X < 4)$

This is the probability that $X$ takes a value less than 4, i.e., the sum of probabilities for $X = 0, 1, 2, 3$:

$P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

Substituting the values in terms of $k$:

$P(X < 4) = k + 3k + 5k + 7k = (1 + 3 + 5 + 7)k = 16k$

Now, substitute $k = \frac{1}{49}$:

$P(X < 4) = 16 \times \frac{1}{49} = \frac{16}{49}$

Step 3: Find $P(3 < X \leq 6)$

This is the probability that $X$ is between 3 and 6, inclusive of 6, i.e., $P(4 \leq X \leq 6)$. This involves summing the probabilities for $X = 4, 5, 6$:

$P(3 < X \leq 6) = P(X = 4) + P(X = 5) + P(X = 6)$

Substituting the values in terms of $k$:

$P(3 < X \leq 6) = 9k + 11k + 13k = (9 + 11 + 13)k = 33k$

Substituting $k = \frac{1}{49}$:

$P(3 < X \leq 6) = 33 \times \frac{1}{49} = \frac{33}{49}$

Final Answers:

• $k = \frac{1}{49}$
• $P(X < 4) = \frac{16}{49}$
• $P(3 < X \leq 6) = \frac{33}{49}$

Would you like further details or explanations on any step? Here are some related questions for further practice:

1. How do you calculate the expected value of $X$ using this PDF?
2. What is the variance of $X$ based on the given probabilities?
3. How do cumulative distribution functions (CDF) relate to PDFs in discrete variables?
4. Can you find $P(X \geq 5)$ from the given PDF?
5. What is the probability that $X$ equals an odd number?

Tip: Always check that the total sum of probabilities equals 1 to confirm that your PDF is valid.