# Can we split a random variable into intervals on its domain of possible values and express it in terms of “simple” distributions on those intervals?

So suppose we have a random variable $Z$ which can take values in $\left[-A, A \right]$. Suppose we do not know the exact distribution of $Z$.

Now if we take $N$ fixed disjoint intervals of $[-A,A]$ such that $\mathop{\dot{\bigcup}}_{i=1}^N Y_i=[-A,A]$ and suppose that we know the probability $p_i=P(Z \in Y_i)$.

Now if we wish to sample $Z$ we can first sample from a categorical distribution to get in which interval $Y_i$ the sample of $Z$ will lie. From there we have to sample from the distribution of $Z|Z \in Y_i$.

If we know the distribution of $Z|Z \in Y_i$ we can easily sample $Z$ in this indirect manner. Now for my question:

Suppose we do not the distribution of $Z|Z \in Y_i$ and approximate this with a uniform distribution. How good is our approximation? Can we get any convergence results if we let $N \rightarrow \infty$ assuming we still can get the $P(Z \in Y_i)$?

Now based on the comments, I see that this is equivalent to asking if we know the cumulative distribution function $F_Z$ at the values $F_Z(x_i)=p_i$ where the $x_i$ correspond to the edges of the intervals $Y_i$ and we do linear interpolation. Now since the cumulative distribution function is a monotonicaly increasing function I think we should be able to achieve some sort of result on how well the linear interpolation approximates this. I hope this clarifies my question.

Futhermore, would it make a difference if $Z$ is unbounded?

## This question has an open bounty worth +100 reputation from Jan ending in 3 days.

This question has not received enough attention.

An answer should include at least an answer to the first question I ask about what we can say about the convergence when using the uniform distribution on these intervals as approximation.

• perhaps, it should be $F_{Y_i}(z) \dots$ – pointguard0 Aug 10 at 14:47
• When you have no information about the $p_i$, it is hard to see how you could proceed. – herb steinberg Aug 10 at 16:22
• Let see if this match with your situation: There is a set of points $\{z_1, z_2, \ldots, z_n\}$ in which you know the corresponding values $F_Z(z_i)$ and you do not know the other points. To approximate by uniform distribution in between is equivalent to drawing a straight line connecting the neighboring points $(z_i, F_Z(z_i)), (z_{i+1}, F_Z(z_{i+1}))$ when you plot the points (assume in ascending order WLOG) as a CDF graph, and you approximate the original CDF by a trapezoid, like the trapezoidal rule in numerical integration. – BGM Aug 11 at 3:11
• can the OP clarify what else is needed? IMHO, @BGM comment pretty much answered the question. if the intervals are picked reasonably, e.g. divide $[-A, A]$ into $N$ equal parts, then as $N \rightarrow \infty$ the piecewise linear approximation will tend to the true CDF, i.e. convergence in distribution en.wikipedia.org/wiki/Convergence_of_random_variables. I am not an expert on the technical details of functional convergence -- is that what the OP requires? Or does the OP want info (e.g. bounds as function of $N$) on some form of en.wikipedia.org/wiki/Statistical_distance ? – antkam yesterday