# Are the following function families the families of the probability densities of some stochastic process?

I am a beginner in stochastic processes, and I am trying to learn this branch of math. I have a few questions about the exercises I solved. I would like to ask if my reasoning was proper, and if the solution is good. The exercise states:

Are the following function families the families of the probability densities of some stochastic process?

(a) $$f_n(\mathbf{t}_n,\mathbf{x}_n)=\left\{\begin{matrix} \frac{1}{t_1t_2\cdots t_n} & for\; 0 \leq x_i \leq t_i,\; i=1,2,...,n\\ 0 & otherwise \end{matrix}\right.$$

(b)$$f_n(\mathbf{t}_n,\mathbf{x}_n)=\left\{\begin{matrix} a_1a_2\cdots a_n\cdot exp(-a_1x_1-a_2x_2...-a_nx_n) & for\; x_1>0,x_2>0,...,x_n>0\\ 0 & otherwise \end{matrix}\right.$$

where $$\mathbf{t}_n=(t_1,t_2,...,t_n)$$, $$\mathbf{x}_n=(x_1,x_2,...,x_n)$$, $$n=1,2,...$$, $$a_1=t_1$$, $$a_i=t_i-t_{i-1}$$.

My solution was to integrate $$f_n(\mathbf{t}_n,\mathbf{x}_n)$$ with respect to some $$x_i$$ and see wether the outcome depends on $$t_i$$. If it does the funcion is not a density, if it doesn't depend on $$t_i$$ it can be a density of some stochastic process.

(a)

$$\int_0^{t_i}f_n(\mathbf{t}_n,\mathbf{x}_n)dx_i = \int_0^{t_i}\frac{1}{t_1 t_2 \cdots t_n}dx_i = \frac{1}{t_1 t_2 \cdots t_n} \int_0^{t_i}dx_i = \frac{x_i|_0^{t_i}}{t_1 t_2 \cdots t_n} = \frac{t_i-0}{t_1 t_2 \cdots t_n} = \frac{1}{t_1 t_2 \cdots t_{i-1}t_{i+1}\cdots t_n}$$

(b)

$$\int_0^{+\infty}f_n(\mathbf{t}_n,\mathbf{x}_n)dx_i= \int_0^{+\infty} a_1a_2\cdots a_n\cdot exp(-a_1x_1-a_2x_2...-a_nx_n)dx_i=$$ $$\prod _{k=1}^na_k\int_0^{+\infty} exp(-a_1x_1-a_2x_2...-a_nx_n)dx_i =$$ $$\prod _{k=1}^na_k \cdot exp(-\sum_{j=1}^{i-1}a_jx_j-\sum_{j=i+1}^na_jx_j)\int_0^{+\infty} exp(-a_ix_i)dx_i =$$ $$\prod _{k=1}^na_k \cdot exp(-\sum_{j=1}^{i-1}a_jx_j-\sum_{j=i+1}^na_jx_j)\frac{1}{-a_i}exp(-a_ix_i)|_0^{+\infty}=$$ $$\prod _{k=1}^na_k \cdot exp(-\sum_{j=1}^{i-1}a_jx_j-\sum_{j=i+1}^na_jx_j)\frac{1}{-a_i}[0-1] =$$ $$\prod _{k=1}^na_k \cdot exp(-\sum_{j=1}^{i-1}a_jx_j-\sum_{j=i+1}^na_jx_j)\frac{1}{a_i}=$$ $$\prod _{k=1\neq i}^na_k \cdot exp(-\sum_{j=1}^{i-1}a_jx_j-\sum_{j=i+1}^na_jx_j)$$

The function given in (a) can be a density, whereas (b) cannot as coefficients $$a_i$$ depend on $$t_{i-1}$$ and $$t_i$$.

Is it good?