The given problem involves a finite sample space $\Omega = \{0, 1, 2, \dots, n\}$ for a positive integer $n$, and a probability function $\mathbb{P}(\{k\}) = c k$ for each $k \in \Omega$, where $c$ is a constant to be determined. You are asked to:

1. Find $c$ in terms of $n$ (1 point).
2. Verify that this satisfies the first and second axioms of probability (2 points).

Step 1: Finding $c$ in terms of $n$

The sum of the probabilities over the entire sample space must be 1 (this is the normalization condition):

$\sum_{k=0}^{n} \mathbb{P}(\{k\}) = 1$

Since $\mathbb{P}(\{k\}) = c k$, this gives:

$\sum_{k=0}^{n} c k = 1$

Factoring out $c$:

$c \sum_{k=0}^{n} k = 1$

The sum of the first $n$ integers is a well-known formula:

$\sum_{k=0}^{n} k = \frac{n(n+1)}{2}$

Substituting this into the equation:

$c \cdot \frac{n(n+1)}{2} = 1$

Solving for $c$:

$c = \frac{2}{n(n+1)}$

Thus, the constant $c$ is:

$c = \frac{2}{n(n+1)}$

Step 2: Verifying the Axioms of Probability

The two fundamental axioms of probability are:

• Axiom 1: The probability of any event is non-negative, i.e., $\mathbb{P}(\{k\}) \geq 0$ for all $k \in \Omega$.
• Axiom 2: The sum of the probabilities of all possible outcomes is 1, i.e., $\sum_{k=0}^{n} \mathbb{P}(\{k\}) = 1$.

Verifying Axiom 1

Since $\mathbb{P}(\{k\}) = c k$, and $k \geq 0$, the probability will always be non-negative as long as $c \geq 0$. From our earlier result, $c = \frac{2}{n(n+1)}$, which is positive for any positive integer $n$. Therefore, the first axiom is satisfied.

Verifying Axiom 2

We already used the second axiom in Step 1 when we summed the probabilities and set it equal to 1. Thus, the second axiom is also satisfied by the calculation:

$\sum_{k=0}^{n} \mathbb{P}(\{k\}) = 1$

Conclusion

• The constant $c$ is $\frac{2}{n(n+1)}$.
• The probability function satisfies both the first and second axioms of probability.

Would you like further clarification or additional details?

Here are five related questions to deepen your understanding:

1. How would the result change if the sample space were modified?
2. What are the implications if $c$ were not positive?
3. Can this probability distribution be generalized to non-uniform cases?
4. How would you handle the case when $n \to \infty$?
5. What if the probability $\mathbb{P}(\{k\})$ had a different dependence on $k$?

Tip: Always check that probability distributions sum to 1, as this is a fundamental requirement in probability theory.