To solve this problem, we need to determine the value of $k$ by using the properties of a probability distribution. The given problem provides a discrete random variable $X$ and its probability distribution $P(X)$.

Key Property:

The sum of all probabilities in a probability distribution must equal 1: $\sum P(X) = 1$

Data from the Table:

The probabilities for each value of $X$ are:

• $P(0) = 0$
• $P(1) = k$
• $P(2) = 2k$
• $P(3) = 2k$
• $P(4) = 3k$
• $P(5) = k^2$
• $P(6) = 2k^2$
• $P(7) = 7k^2 + k$

Step 1: Write the Total Probability Equation

$P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7) = 1$ Substitute the probabilities: $0 + k + 2k + 2k + 3k + k^2 + 2k^2 + (7k^2 + k) = 1$

Step 2: Combine Like Terms

Combine all $k$ terms and $k^2$ terms: $k + 2k + 2k + 3k + k + 7k = 16k$ $k^2 + 2k^2 + 7k^2 = 10k^2$ Thus, the equation becomes: $16k + 10k^2 = 1$

Step 3: Solve the Quadratic Equation

Rearrange the equation: $10k^2 + 16k - 1 = 0$

Use the quadratic formula to solve for $k$: $k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 10$, $b = 16$, and $c = -1$. Substituting these values: $k = \frac{-16 \pm \sqrt{16^2 - 4(10)(-1)}}{2(10)}$ $k = \frac{-16 \pm \sqrt{256 + 40}}{20}$ $k = \frac{-16 \pm \sqrt{296}}{20}$ $k = \frac{-16 \pm 2\sqrt{74}}{20}$ $k = \frac{-8 \pm \sqrt{74}}{10}$

Step 4: Select the Positive Solution

Since $k$ represents a probability scaling factor, it must be positive. Thus: $k = \frac{-8 + \sqrt{74}}{10}$

This is the final value of $k$.

Do you want me to simplify further, explain this in detail, or confirm with calculations? Let me know!

Here are 5 related questions for practice:

1. What happens if one of the probabilities is negative?
2. How can you verify if a given distribution is valid?
3. How does the quadratic formula work in solving such problems?
4. What is the significance of each $k^2$ term in the distribution?
5. How would the solution change if additional constraints were given for $k$?

Tip: Always check that all probabilities sum to 1 and are non-negative in any discrete probability distribution.