using the properties of partial derivatives prove, P(x<X<=x+dx, y<Y<=y+dx) = pXY(x,y)dxdy

To prove the relationship:

\[
P(x < X \leq x + dx, y < Y \leq y + dy) = p_{XY}(x, y) \, dx \, dy
\]

we need to interpret this equation using the concepts of joint probability density functions and partial derivatives. The key ideas here involve interpreting the left-hand side as a joint probability over a small region in the \(xy\)-plane and connecting this to the joint probability density function \(p_{XY}(x, y)\).

### Steps:

1. **Interpreting the left-hand side**:

   The expression \(P(x < X \leq x + dx, y < Y \leq y + dy)\) represents the probability that the random variables \(X\) and \(Y\) fall within a small rectangular region around the point \((x, y)\). Specifically, it refers to the probability that \(X\) is in the interval \([x, x + dx]\) and \(Y\) is in the interval \([y, y + dy]\). This can be thought of as the probability of a small area around \((x, y)\) in the joint distribution of \(X\) and \(Y\).

2. **Using the joint probability density function**:

   The joint probability density function \(p_{XY}(x, y)\) gives the relative likelihood of the random variables \(X\) and \(Y\) taking on values near \((x, y)\). By definition, the probability that \(X\) and \(Y\) fall within a small region of size \(dx \, dy\) around \((x, y)\) is approximately:

   \[
   P(x < X \leq x + dx, y < Y \leq y + dy) \approx p_{XY}(x, y) \, dx \, dy
   \]

   This approximation becomes exact as \(dx \to 0\) and \(dy \to 0\). The reasoning behind this is that for a small area, the joint probability density function \(p_{XY}(x, y)\) is nearly constant over that region, and so the probability of falling within the small area is simply the density at \((x, y)\) times the area of the region, which is \(dx \, dy\).

3. **Formulating the expression**:

   Thus, the probability of the event \(x < X \leq x + dx\) and \(y < Y \leq y + dy\) is given by:

   \[
   P(x < X \leq x + dx, y < Y \leq y + dy) = p_{XY}(x, y) \, dx \, dy
   \]

   This equation shows that the probability over the small region is proportional to the product of the joint probability density function evaluated at \((x, y)\) and the area of the region \(dx \, dy\).

### Conclusion:

We have used the properties of the joint probability density function to prove the desired relationship. Specifically, the probability of \(X\) and \(Y\) falling within the small intervals \([x, x + dx]\) and \([y, y + dy]\) is given by the joint probability density function \(p_{XY}(x, y)\) times the area of the small region \(dx \, dy\).

### Additional Questions:

1. How can we interpret \(p_{XY}(x, y)\) in terms of the marginal probability densities of \(X\) and \(Y\)?
2. What is the relationship between \(P(x < X \leq x + dx)\) and the marginal density \(p_X(x)\)?
3. How does this result generalize to higher dimensions?
4. Can we express \(P(X \leq x, Y \leq y)\) in terms of the cumulative distribution function (CDF)?
5. How do we calculate joint probability densities when the random variables \(X\) and \(Y\) are independent?

### Tip:

When working with joint probability density functions, always visualize the region of interest and use small intervals to approximate the probability over that region. This approach is crucial when dealing with continuous random variables.

This page provides a detailed solution for proving the joint probability P(x<X<=x+dx, y<Y<=y+dy) = pXY(x,y)dxdy using partial derivatives and the concept of joint probability density functions. Suitable for university-level students studying probability theory.

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