Given a Beta Random Variable $X$ with parameters $\alpha, \beta$ and a positive constant $n$, suppose I am interested in the distribution of:
$$Y:=\lfloor nX\rfloor$$
Suppose I want a histogram showing the distribution of $Y$ in Mathematica. How can I go about plotting this histogram? Is it necessary that I generate many samples to approximate it first, or can Mathematica calculate it perfectly? Would you please provide some example code showing how you can obtain and plot this distribution?
If you are interested in the distribution of $Y$, you don't want a histogram of counts. $Y$ is a discrete random variable. You want the vertical axis to be the estimated probability for the values of $Y$.
A DiscretePlot is what you want. One way to get the appropriate plot is to use HistogramList to get the probabilities.
data = Floor /@ (20 RandomVariate[BetaDistribution[4, 3], 1000]);
probability = HistogramList[data, {Min[data] - 1/2, Max[data] + 1/2, 1}, "PDF"]
(* {{3/2, 5/2, 7/2, 9/2, 11/2, 13/2, 15/2, 17/2, 19/2, 21/2, 23/2, 25/2,
27/2, 29/2, 31/2, 33/2}, {1/500, 13/500, 21/1000, 51/1000, 29/500,
23/250, 93/1000, 1/8, 97/1000, 57/500, 19/200, 9/100, 31/500, 57/1000, 17/1000}} *)
DiscretePlot[probability[[2, i - Min[data] + 1]], {i, Min[data], Max[data]}]
If you really have to have something that looks like a histogram (like if you boss insists on it or you're stuck in the 20th century), then you need to make sure that the bars are centered on the integer values:
Histogram[data, {Min[data] - 1/2, Max[data] + 1/2, 1}, "PDF"]
If you don't include {Min[data] - 1/2, Max[data] + 1/2, 1}, then the default will have the bars centered on 0.5, 1.5, 2.5, etc., which are values that $Y$ can't take on.
ClearAll[td]
td[n_, α_, β_] := TransformedDistribution[Floor[n x],
Distributed[x, BetaDistribution[α, β]]]
sample = RandomVariate[td[10, 2, 4], 500] ;
Histogram[sample]
Histogram[sample, Automatic, "PDF"]
Expectation[x, x \[Distributed] td[10, 2, 4]]
5667/2000
Histogram[Floor /@ (20 RandomVariate[BetaDistribution[4, 3], 1000])]
