#StackBounty: #self-study #stochastic-processes Are the following function families the families of the probability densities of some s…

Bounty: 100

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) $$f_n(mathbf{t}_n,mathbf{x}_n)=left{begin{matrix}
frac{1}{t_1t_2cdots t_n} & for; 0 leq x_i leq t_i,; i=1,2,…,n\
0 & otherwise
end{matrix}right.$$

(b)$$f_n(mathbf{t}_n,mathbf{x}_n)=left{begin{matrix}
a_1a_2cdots a_ncdot exp(-a_1x_1-a_2x_2…-a_nx_n) & for; x_1>0,x_2>0,…,x_n>0\
0 & otherwise
end{matrix}right.$$

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)

$$int_0^{+infty}f_n(mathbf{t}n,mathbf{x}_n)dx_i= int_0^{+infty} a_1a_2cdots a_ncdot exp(-a_1x_1-a_2x_2…-a_nx_n)dx_i=$$
$$prod _{k=1}^na_kint_0^{+infty} exp(-a_1x_1-a_2x_2…-a_nx_n)dx_i =$$
$$ prod _{k=1}^na_k cdot exp(-sum
{j=1}^{i-1}a_jx_j-sum_{j=i+1}^na_jx_j)int_0^{+infty} exp(-a_ix_i)dx_i =$$
$$prod {k=1}^na_k cdot exp(-sum{j=1}^{i-1}a_jx_j-sum_{j=i+1}^na_jx_j)frac{1}{-a_i}exp(-a_ix_i)|0^{+infty}=$$
$$prod _{k=1}^na_k cdot exp(-sum
{j=1}^{i-1}a_jx_j-sum_{j=i+1}^na_jx_j)frac{1}{-a_i}[0-1] =$$
$$prod {k=1}^na_k cdot exp(-sum{j=1}^{i-1}a_jx_j-sum_{j=i+1}^na_jx_j)frac{1}{a_i}= $$
$$prod {k=1neq i}^na_k cdot exp(-sum{j=1}^{i-1}a_jx_j-sum_{j=i+1}^na_jx_j)$$

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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