Do you know any results or articles about change money? Something like the statistics of different value notes in a cash box. Or answers to questions which distribution of notes values is best for starting a day in a shop. I mean obviously you need more small value notes than large ones. After one day of selling you probably have more large notes as they don't go away easily.
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Regarding the comment (since I have insufficient reputation to comment), "change money" is the cash kept in a drawer at the start of business in a shop. You will want to have some money on hand, so that if someone comes in with, say, a \$20 bill, you can give them appropriate change for a \$17 purchase.
To answer the question, you can fairly easily derive appropriate statistics. Without loss of generality, assume that all transactions are whole-dollar transactions.
First, you need to know what the available denominations are. Then, you must estimate the different "tender" amounts -- i.e. the cash a customer gives you before you give them change. You don't need to estimate the distribution of their denominations: \$45 given for a \$41 purchase is identical whether it is given as 2 \$20s and 1 \$5, or 4 \$10s and 1 \$5.
For a reasonable range of these values, estimate what your returned change would be for purchase values up to that amount minus \$1.
If the customer gives you \$20, for the following purchases you need the following bills:
\$1 purchase: 1 \$10, 1 \$5, 4 \$1
\$2 purchase: 1 \$10, 1 \$5, 3 \$1 ...
Some patterns will obviously repeat here.
Because it is straightforward to compute this (this is just doing mixed-base arithmetic), the only interesting part comes from estimating the histogram of the purchase amount. This will vary based on your business; a cafe will have different values than a bookstore, for instance.
Therefore, you have one random variable, the purchase amount, which can be easily associate with different change amounts for different "tender" values. The "tender" value is not really a random variable, because the customer almost always gives the smallest denomination that fits a purchase.
Alternatively, you could consider the delta -- the value in change -- as the random variable, which will have a deterministic "change value" associated with it. Again, this is just converting some random number with mixed-base arithmetic.
Coin production figures by the US Mint are available here. I presume coin figures in circulation and in change trays would be in similar proportions. No doubt figures for bills and for other countries are also available to those willing to type a few cleverly chosen words into the web.