Syntax for Nested Functions

Questions
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1

I’m hoping this is a relatively simple question, but I’m quite new to Mathematica and am struggling with exactly how to express what I’d like it to do. The following isn’t what I’m trying to do, but a very simplified example with the same fundamental issue.

We want to calculate the total weight of a collection of balls 1, 2,…,N. There are only a certain number of types of ball, 1, 2,…T. Ball number x is of type balltype[x]. The weight of a ball of type x is weight[x]. So the weight of ball number x is expressed by weight[balltype[x]]. I want to define the function that is the total weight of all N balls, and it should be something like:

totalweight[weight_, balltype_, N_] := Sum[weight[balltype[x]], {x, 1, N}]

The problem is that I’m not now sure how then to ask for the derivative of totalweight with respect to the weight of a specific type of ball, say weight[1]. The answer should be something like

Sum[Boole[balltype[x]=1],{x,1,N}]

i.e. the total number of balls of type 1 that are in the collection, although there are may be many ways to express this.

My question is: how can I express the function totalweight properly, and how (for example) would I ask for the derivative of totalweight wrt the weight of a particular ball type weight[x]? This seems like a simple sort of problem, but I can’t seem to find anything addressing it, although that could be because I don’t know the right terms to search for. Very many thanks in advance.

2

To understand your problem, I made small example and set up a function to find the total weight of all balls of a certain type:

In[3]:= SeedRandom[1234](*10 ball types with a different weight each \
between 10 and 50*)
ballWeightsByType = RandomReal[{10, 50}, 10]

Out[4]= {45.0643, 30.8786, 13.4489, 25.1165, 10.4658, 47.0906, \
31.7503, 29.1733, 19.814, 40.3958}

In[5]:= (*random collection of 50 balls identified by their ball type*)

In[6]:= ballsByType = RandomInteger[{1, 10}, 50]

Out[6]= {3, 9, 9, 1, 5, 8, 4, 7, 6, 4, 8, 10, 1, 5, 5, 3, 3, 5, 8, 6, \
9, 3, 3, 7, 8, 3, 6, 3, 1, 3, 8, 10, 6, 7, 8, 1, 10, 6, 8, 8, 8, 10, \
8, 8, 6, 1, 6, 5, 4, 10}

In[7]:= (*get weight of balltype x*)

In[8]:= ballWeight[x_?IntegerQ] := First@Take[ballWeightsByType, {x}]

In[9]:= ballWeight[7]

Out[9]= 31.7503

In[10]:= (*list of {type,weight} of all 50 balls sorted by type*)

In[11]:= lst = SortBy[{#, ballWeight[#]} & /@ ballsByType, First]

Out[11]= {{1, 45.0643}, {1, 45.0643}, {1, 45.0643}, {1, 45.0643}, {1, 
  45.0643}, {3, 13.4489}, {3, 13.4489}, {3, 13.4489}, {3, 
  13.4489}, {3, 13.4489}, {3, 13.4489}, {3, 13.4489}, {3, 
  13.4489}, {4, 25.1165}, {4, 25.1165}, {4, 25.1165}, {5, 
  10.4658}, {5, 10.4658}, {5, 10.4658}, {5, 10.4658}, {5, 
  10.4658}, {6, 47.0906}, {6, 47.0906}, {6, 47.0906}, {6, 
  47.0906}, {6, 47.0906}, {6, 47.0906}, {6, 47.0906}, {7, 
  31.7503}, {7, 31.7503}, {7, 31.7503}, {8, 29.1733}, {8, 
  29.1733}, {8, 29.1733}, {8, 29.1733}, {8, 29.1733}, {8, 
  29.1733}, {8, 29.1733}, {8, 29.1733}, {8, 29.1733}, {8, 
  29.1733}, {8, 29.1733}, {9, 19.814}, {9, 19.814}, {9, 19.814}, {10, 
  40.3958}, {10, 40.3958}, {10, 40.3958}, {10, 40.3958}, {10, 40.3958}}

In[12]:= (*number of balls of type 7*)

In[13]:= Count[lst, {7, _}]

Out[13]= 3

In[14]:= (*weight of all balls of type 6*)

In[15]:= % ballWeight[7]

Out[15]= 31.7503 Null

In[16]:= weightTotalByType[type_?IntegerQ] := 
 Count[lst, {type, _}]*ballWeight[type]

In[17]:= weightTotalByType[7]

Out[17]= 95.2508

In[18]:= (*list of {type,total weight of all balls of type} of all 50 \
balls sorted by type*)

In[19]:= {#, weightTotalByType[#]} & /@ Range[10]

Out[19]= {{1, 225.322}, {2, 0.}, {3, 107.591}, {4, 75.3496}, {5, 
  52.3289}, {6, 329.635}, {7, 95.2508}, {8, 320.906}, {9, 
  59.4419}, {10, 201.979}}
3

Many thanks for this - yes, this is exactly the sort of problem I am thinking about. However, I don’t want to analyse a specific set of weights and types of ball. I want to leave them as free variables - weight[1], balltype[3] etc., and then manipulate the ‘totalweight’ function using those variables, to (for example) calculate the derivative of totalweight with respect to the weight of a specific ball type (e.g. weight[1]). I am trying to do this using statements like

D[totalweight[weight, balltype, N], weight[1]]

but I’m certain this is not well-formed, and it returns the output ‘0’.

(The problem I’m actually looking at is to derive maximisation conditions for a big PDF, which expresses the outcome of a series of experiments, where each experiment has an individual PDF depending on its ‘type’ - but I didn’t want to complicate things when the problem I’ve got is basically a syntactical one.)