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.


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


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

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?


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