Let's analyze and solve the problem step by step:

Problem Statement:

The table gives the probability distribution of a random variable $X$, with the probability function values $P(X = x)$ listed for $x = 0, 1, 2, 3, 4, 5, 6, 7$. The values of $P(X = x)$ depend on the constant $k$, and we are tasked with:

1. Finding $k$.
2. Evaluating $P(X < 6)$ and $P(0 < X < 5)$.
3. Determining the cumulative distribution function (CDF) $F(x)$.

Step 1: Find $k$

The sum of all probabilities in a probability distribution must equal 1: $P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) = 1.$

Substitute the given probabilities: $k + k + 2k + 2k + 3k + k^2 + 2k^2 + 7k^2 + k = 1.$

Combine like terms: $5k + 10k^2 = 1.$

Divide through by $k$ (assuming $k \neq 0$): $5 + 10k = \frac{1}{k}.$

Multiply through by $k$: $5k + 10k^2 = 1.$

Rearrange into standard quadratic form: $10k^2 + 5k - 1 = 0.$

Solve this quadratic equation using the quadratic formula: $k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$ where $a = 10$, $b = 5$, $c = -1$.

Substitute: $k = \frac{-5 \pm \sqrt{5^2 - 4(10)(-1)}}{2(10)}.$ $k = \frac{-5 \pm \sqrt{25 + 40}}{20}.$ $k = \frac{-5 \pm \sqrt{65}}{20}.$

Take the positive root (since $k > 0$): $k = \frac{-5 + \sqrt{65}}{20}.$

**Step 2: Evaluate $P(X < 6)$ and $P(0 < X < 5)$

(a) $P(X < 6)$:

This means summing probabilities for $X = 0, 1, 2, 3, 4, 5$: $P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5).$ $P(X < 6) = k + k + 2k + 2k + 3k + k^2 = 9k + k^2.$

(b) $P(0 < X < 5)$:

This means summing probabilities for $X = 1, 2, 3, 4$: $P(0 < X < 5) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).$ $P(0 < X < 5) = k + 2k + 2k + 3k = 8k.$

Step 3: Determine the Cumulative Distribution Function (CDF) $F(x)$

The CDF $F(x)$ is defined as: $F(x) = P(X \leq x).$

Compute $F(x)$ step by step for each $x$:

• $F(0) = P(X = 0) = k.$
• $F(1) = P(X = 0) + P(X = 1) = k + k = 2k.$
• $F(2) = F(1) + P(X = 2) = 2k + 2k = 4k.$
• $F(3) = F(2) + P(X = 3) = 4k + 2k = 6k.$
• $F(4) = F(3) + P(X = 4) = 6k + 3k = 9k.$
• $F(5) = F(4) + P(X = 5) = 9k + k^2.$
• $F(6) = F(5) + P(X = 6) = 9k + k^2 + 2k^2 = 9k + 3k^2.$
• $F(7) = F(6) + P(X = 7) = 9k + 3k^2 + 7k^2 = 9k + 10k^2.$

Since $F(7) = 1$, this serves as a verification for the value of $k$.

Would you like me to substitute the value of $k$ back into these expressions and compute explicit numerical results?