Let $X=(X_1,..,X_n)$ be a density random vector.
Then for all $1 \le k \le n$: $$f_{X_k}(x_k)=\int_{-\infty}^{\infty}dx_1···\int_{-\infty}^{\infty}dx_{k-1}\int_{-\infty}^{\infty}dx_{k+1}···\int_{-\infty}^{\infty}dx_n f_X(x_1,..,x_n)$$
Proof:
$$P(X_k\in A)=P(X\in\mathbb{R}^{k-1}\times A \times \mathbb{R}^{n-k})=$$$$\int_{\mathbb{R}}dx_1···\int_{\mathbb{R}}dx_{k-1}\int_{A}dx_k\int_{\mathbb{R}}dx_{k+1}\int_{\mathbb{R}}dx_{n}f_X(x_1,..,x_n)=$$$$\int_{A}dx_kf_{X_k}(x_k)$$
I don't understand the last two equalities, is it by definition that $P(X_1\le x_1, …,X_n \le x_n)=\int_{-\infty}^{x_1}ds_1···\int_{-\infty}^{x_n}ds_nf_X(s_1,..,s_n)$ ?
And how to explain the last equality ?
Best Answer
It is by definition that $$P((X_1,\dots,X_n)\in B)=\int\;d(x_1,\dots,x_n)\mathbf1_B(x_1,\dots,x_n)f_X(x_1,\dots,x_n)$$ or in slightly different notation:$$P((X_1,\dots,X_n)\in B)=\int_B\;d(x_1,\dots,x_n)f_X(x_1,\dots,x_n)$$
As a sidenote: personally I dislike this notation and would rather go for $$P((X_1,\dots,X_n)\in B)=\int\dots\int\mathbf1_B(x_1,\dots,x_n)f_X(x_1,\dots,x_n)\;dx_1\dots dx_n$$
In special case $B=A_1\times\cdots\cdots\times A_n$ this takes the form:$$P(X_1\in A_1,\dots,X_n\in A_n)=\int_{A_1}\;dx_1\cdots\int_{A_n}\;dx_nf_X(x_1,\dots,x_n)$$
Now see what happens if $A_i=(-\infty,x_i]$.
Concerning the second equality note that it is allowed here to change the order of integration.
So: $$\int_{\mathbb R}dx_1\cdots\int_Adx_k\cdots\int_{\mathbb R}dx_nf_X(x_1,\dots,x_n)=\int_{A}dx_k\int_{\mathbb R}dx_1\cdots\int_{\mathbb R}dx_nf_X(x_1,\dots,x_n)=\int_Adx_kf_{X_k}(x_k)$$