Let A and B be events and let X and Y be the Bernoulli
random variables corresponding to A and B (
X =
(
1, if the outcome is in A,
0, if the outcome is in Ac
). Show that A and B are 
independent (P rob(A and B) = P rob(A) × P rob(B)) if and only if X and Y are independent  E(XnY
m) = E(Xn
) E(Y
m)

### To show that \( A \) and \( B \) are independent if and only if the Bernoulli random variables \( X \) and \( Y \) corresponding to these events are independent, we need to approach the problem in two parts.

**1. Showing that \( A \) and \( B \) are independent implies \( X \) and \( Y \) are independent.**

#### Assumptions:
- \( A \) and \( B \) are independent events, which means:
  \[
  P(A \cap B) = P(A) \cdot P(B)
  \]

#### Corresponding Bernoulli random variables:
- Let \( X \) and \( Y \) be the Bernoulli random variables for events \( A \) and \( B \) respectively:
  \[
  X =
  \begin{cases}
  1, & \text{if the outcome is in } A \\
  0, & \text{if the outcome is in } A^c
  \end{cases}
  \]
  \[
  Y =
  \begin{cases}
  1, & \text{if the outcome is in } B \\
  0, & \text{if the outcome is in } B^c
  \end{cases}
  \]

#### Objective:
We need to show that \( E(X^n Y^m) = E(X^n) \cdot E(Y^m) \) for all non-negative integers \( n \) and \( m \) when \( A \) and \( B \) are independent.

#### Proof:
Since \( X \) and \( Y \) are Bernoulli random variables, their powers are simple:
- \( X^n = X \) for any \( n \geq 1 \) (since \( X \in \{0, 1\} \)),
- \( Y^m = Y \) for any \( m \geq 1 \).

Thus, for any \( n \) and \( m \), we are really proving:
\[
E(XY) = E(X) \cdot E(Y)
\]

Now, compute each term:
- \( E(X) = P(A) \) (since \( X = 1 \) when the outcome is in \( A \), and 0 otherwise),
- \( E(Y) = P(B) \),
- \( E(XY) = P(A \cap B) \) (since \( XY = 1 \) only if both \( X = 1 \) and \( Y = 1 \), which happens if and only if the outcome is in both \( A \) and \( B \)).

Since \( A \) and \( B \) are independent, we have:
\[
P(A \cap B) = P(A) \cdot P(B)
\]
Thus:
\[
E(XY) = P(A \cap B) = P(A) \cdot P(B) = E(X) \cdot E(Y)
\]
Therefore, \( E(XY) = E(X) \cdot E(Y) \), which shows that \( X \) and \( Y \) are independent.

---

**2. Showing that \( X \) and \( Y \) are independent implies \( A \) and \( B \) are independent.**

#### Assumptions:
- \( X \) and \( Y \) are independent random variables, which means:
  \[
  E(XY) = E(X) \cdot E(Y)
  \]

#### Objective:
We need to show that this implies:
\[
P(A \cap B) = P(A) \cdot P(B)
\]

#### Proof:
As before, we know that:
- \( E(X) = P(A) \),
- \( E(Y) = P(B) \),
- \( E(XY) = P(A \cap B) \).

Since \( X \) and \( Y \) are independent, we are given:
\[
E(XY) = E(X) \cdot E(Y)
\]
Substituting the expressions for \( E(X) \) and \( E(Y) \), we get:
\[
P(A \cap B) = P(A) \cdot P(B)
\]
This proves that \( A \) and \( B \) are independent.

---

### Conclusion:
We have shown that \( A \) and \( B \) are independent if and only if the corresponding Bernoulli random variables \( X \) and \( Y \) are independent, which completes the proof.

### Follow-up questions:
1. How do Bernoulli random variables differ from general random variables?
2. What happens if \( A \) and \( B \) are dependent events? How does that affect the proof?
3. Can this result be generalized to non-Bernoulli random variables?
4. How can we extend this proof to more than two events?
5. What is the significance of the expectation operator in proving independence?

### Tip:
When dealing with Bernoulli random variables, remember that their powers are either 0 or 1, which simplifies many calculations involving products and expectations.

Let A and B be events and let X and Y be the Bernoulli random variables corresponding to A and B (X = (1, if the outcome is in A, 0, if the outcome is in Ac)). Show that A and B are independent (P(A and B) = P(A) × P(B)) if and only if X and Y are independent (E(X^n Y^m) = E(X^n) E(Y^m)).

This page demonstrates how to prove that two events A and B are independent if and only if their corresponding Bernoulli random variables X and Y are independent. The proof involves key probability concepts, including the expectation of Bernoulli variables and the formula for the independence of events.

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