Here is a breakdown of the questions from the image:

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 $x$.

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.

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.