Thanks, Jim Baldwin. That works and it's straightforward. But it doesn't address the difficulty I'm focused on. I can center the bins over the numbers the way you showed. Or I can set the bin-width. My problem is that I cannot do both at the same time.

Thanks again, Jim Baldwin, for your commitment to my question. Code below should illustrate my problem. I want to accomplish both proper alignment and separated bars in one histogram. Others must have faced the issue when illustrating discrete sets. I can accomplish this, but not in any straightforward way that would make sense to anyone including to me.

poisson = RandomVariate[PoissonDistribution[1], 500];
Histogram[poisson, {1/4}, 
 PlotLabel -> "Bars Separated. Numbers Misaligned.", 
 AspectRatio -> 1/3]
Histogram[poisson, {-1/2, 6, 1}, 
 PlotLabel -> "Numbers Aligned. Bars Not Separated.", 
 AspectRatio -> 1/3]

I appreciate your interest. This is an assignment in a physics class on using computers for statistical analysis. This image is part of one problem focusing on the Central Limit Theorem. Its purpose is to illustrate that Poisson is a skewed distribution of non-negative integers. I plan to adopt your idea to use DiscretePlot. That better illustrates the point I want to make.

The problem is part of a much larger assignment. I pulled it out of a draft I'm still working on and published it at Baldwin Conversation . It doesn't include the explanation of shading. Light brown is code. My narrative is light blue. The unshaded portions are from the assignment itself.

I'm the only math major and the only undergraduate in the class. I'm not proficient with Mathematica, and I'm using it under a dispensation from the professor. So I want the images to shine.