Integral expression for $P(X_1 \leq x_1, ...., X_n \leq x_n \mid N = n)$

Question

What is the integral expression for $P(X_1 \leq x_1, ...., X_n \leq x_n\mid N = n)$? I am confused by having a joint and conditional probability.

I know: $P(X_1 \leq x_1 , ..., X_n \leq x_n) = \int_{-\infty}^{x_1} ... \int_{-\infty}^{x_n} f(u_1,...,u_n) du_1 ... du_n$

And I also thought to use: $P(X=x,Y=y)=P(Y=y)P(X=x\mid Y=y)$

But I don't know how to connect these. Can somebody explain? Thanks in advance!

Answer 1

Edit: As the comment eludes to, having $X_n$ and $N$ is confusing. Suppose we have a random variable $Y$ in place of $N$.

I assume you are dealing with continuous random variables. Let $X = (X_1, \dots, X_n)$ and $x = (x_1, \dots, x_n)$. Then the conditional cdf of $X$ given $Y = y$ is defined as follows

$$ \begin{align} F_{X|Y}(x|y) &= P(X \leq x | Y = y) \\ &= \int_{-\infty}^{x_1} \cdots \int_{-\infty}^{x_n}f_{X|Y}(x|y)dx \\ &= \frac{1}{f_Y(y)}\int_{-\infty}^{x_1} \cdots \int_{-\infty}^{x_n} f_{XY}(x,y)dx. \end{align} $$

Here, $f_{X|Y}(x|y) = \frac{f_{XY}(x,y)}{f_Y(y)}$.