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**2. Define conditional probability \( P(A|B) \), where \( P(B) > 0 \). If \( P(A^C \cap B) = 0.4 \) and \( P(B) = 0.6 \), compute the conditional probability \( P(A|B) \). Define a distribution function of a random variable.**

### Solution:
The formula for conditional probability is:

\[
P(A|B) = \frac{P(A \cap B)}{P(B)}
\]

We are given:
- \( P(A^C \cap B) = 0.4 \) (This means the probability of the complement of \( A \) and \( B \)),
- \( P(B) = 0.6 \).

Since \( P(A^C \cap B) + P(A \cap B) = P(B) \), we can find \( P(A \cap B) \):

\[
P(A \cap B) = P(B) - P(A^C \cap B) = 0.6 - 0.4 = 0.2
\]

Now, calculate \( P(A|B) \):

\[
P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.6} = \frac{1}{3} \approx 0.333
\]

**Definition of a distribution function**:
A distribution function, also known as a cumulative distribution function (CDF), of a random variable \( X \) is defined as:

\[
F_X(x) = P(X \leq x)
\]

This function gives the probability that the random variable \( X \) takes on a value less than or equal to \( x \).

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**3. Give an example of the distribution function of a probability distribution which is neither discrete nor continuous. Define discrete and continuous random variables.**

### Solution:
An example of a distribution that is neither purely discrete nor continuous is a **mixed distribution**. In a mixed distribution, part of the probability mass is concentrated on specific points (discrete), while the rest is spread continuously over an interval. An example is:

- A distribution where 50% of the probability is assigned to a point \( x_0 \), and the other 50% is spread uniformly over an interval \( [a, b] \).

#### Definitions:
- **Discrete random variable**: A random variable that takes on a finite or countable number of values. For example, the number of heads in coin flips.
- **Continuous random variable**: A random variable that takes values from a continuous range, meaning it can take any value within an interval. For example, the time it takes for a light bulb to burn out, measured in hours.

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Would you like more details or explanations on these concepts?

Here are five related questions to deepen your understanding:
1. What is the relationship between conditional probability and independent events?
2. How do you compute \( P(A|B) \) when given a table of probabilities?
3. What are some real-world examples of mixed probability distributions?
4. How do you distinguish between a CDF and a probability density function (PDF)?
5. Can a random variable be partially discrete and partially continuous in real-life situations?

**Tip**: When working with conditional probabilities, always make sure that the event you are conditioning on (in this case, \( B \)) has a non-zero probability, otherwise the conditional probability is undefined.

2. Define conditional probability P(A|B), where P(B) > 0. If P(A^C ∩ B) = 0.4 and P(B) = 0.6, compute the conditional probability P(A|B). Define a distribution function of a random variable.

3. Give example of the distribution function of a probability distribution which is neither discrete nor continuous. Define discrete and continuous random variables.

Learn how to compute conditional probability P(A|B) and explore examples of mixed probability distributions, including definitions of discrete and continuous random variables. This problem is suitable for undergraduate students or advanced high school learners.

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