[?] Min and Max value (range) of a variate

Questions
1

Hi,

If I have a distribution, e.g.:

p := DiscreteUniformDistribution[{1,6}]

is there any way to find out range of values of related variate? (e.g. [1,6] in my example)
Thank you

2

Use Quantile for that:

Quantile[p, {0, 1}]
3

Thank you! I would have never found it on my own. I was using this:

max = Maximize[{x, Probability[v ==x, v\[Distributed]w] > 0}, x, Integers]
min = Minimize[{x, Probability[v ==x, v\[Distributed]w] > 0}, x, Integers]

I noticed interesting thing, though – Quantile[p, {0, 1}] can return wider range than min/max above if distribution p was defined via EmpiricalDistribution with some zero entries. E.g. this:

p := DiscreteUniformDistribution[{1,6}]

fixd[dist_] := Module[{x,k}, 
  EmpiricalDistribution[ Table[Probability[x==k, x\[Distributed]dist], {k,2,40}]->Table[k, {k, 2, 40}] ]
]

y := fixd[ TransformedDistribution[a + b, {a\[Distributed]p, b\[Distributed]p}] ]

max = Maximize[{x, Probability[v ==x, v\[Distributed]y] > 0}, x, Integers]
min = Minimize[{x, Probability[v ==x, v\[Distributed]y] > 0}, x, Integers]
Quantile[w, {0, 1}]

results:

Out[5]={12,{x->12}}
Out[6]={2,{x->2}}
Out[8]={2,40}

Is it normal?

4

Doesn’t work very well with TransformedDistribution unfortunately:

p := DiscreteUniformDistribution[{1,6}]
p1 := TransformedDistribution[a + b, {a\[Distributed]p, b\[Distributed]p}]
Quantile[p1, {0, 1}]

result:

Out[3]={-?,12}
5

In principle it is not wrong, as Quantile from a UniformDistribution can be the start but also the values before (the CDF is constant there, and thus the value returned by Quantile can be different ones)… to find the ‘maximum’ possible one, I guess you do need to do it manually…

Another way would be to extract it from:

CDF[p1, x]

or

PDF[p1,x]

But it depends on how complicated it is…

e.g.:

p:=DiscreteUniformDistribution[{1,6}]
p1:=TransformedDistribution[a+b,{a\[Distributed]p,b\[Distributed]p}]
cdf=CDF[p1,x]
MinMax[IntervalUnion@@Cases[cdf,x_<= _<y_:>Interval[{x,y}],\[Infinity]]]
6

Some slightly hacky workarounds, which work since p1 is discrete:

Quantile[p1, {$MinMachineNumber, 1}]
Quantile[p1, {$MachineEpsilon, 1}]
Quantile[p1, {1/10^1000, 1}]

One can see the problem by evaluating InverseCDF[p1, x]. And you can use InverseCDF[p1, {low, high}], where low is greater than zero but small enough yield the lowest value in the range (similarly for high):

InverseCDF[p1, {$MachineEpsilon, 1}]