How do I overlap a Poisson distribution with a histogram 
I expect to get a Poisson distribution over my histogram but as I can see from the graph, I get a straight line. Would really appreciate insight.
 A: In order to get a reasonable match between the
histogram of a sample and the PDF of the population
you will likely need a sample of several thousand.
Also, you need to make a 'probability' histogram in
which the sum of the areas of the bars is unity.
Below I set the bin boundaries to be half integers
so that each bar represents only one possible Poisson
value.
Finally, using a density histogram allows you to
plot the density function of the approximating
normal distribution on the same scale.
Here is a plot from R, using standard graphics from
the core of R.
Sample from $\mathsf{Pois}(\lambda=85)$ and summary:
set.seed(2022)  # for reproducibility 
x = rpois(10009, 85)
summary(x)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  51.00   79.00   85.00   84.92   91.00  120.00 

Histogram with the PDF of $\mathsf{Pois}(\lambda = 85)$
[centers of open red circles], and the density
function of $\mathsf{Norm}(\mu=\lambda, \sigma=\sqrt{\lambda}).$
cutp = (30:155) + .5
hdr = "10,000 Observations from POIS(85), with PDF and Normal Fit"
hist(x, prob=T, br=cutp, col="skyblue2", main=hdr)

k = 55:115
pdf = dpois(k,85)
points(k, pdf, col="red")

curve(dnorm(x,85,sqrt(85)), add=T


