Density and Distribution Functions: Two Ways to Describe One

A distribution function reports the probability that a quantity lands at or below a stated value, and every distribution has one. A density is the rate at which that probability accumulates, so it exists only where the distribution function is smooth and it is never itself a probability. A state price density is that same rate carrying a discount, so it totals the price of a certain rupee rather than one.

A measure assigns a number to a set. Both objects in this guide are ways of writing that assignment down along the number line, and they differ only in how they write it. The distribution function does it directly, one value at a time, by naming how much of the total sits at or below that value. The density does it by rate, and a rate has to be multiplied by a width before it becomes a quantity at all. Everything that follows is a consequence of that single difference, including the awkward one: only one of the two exists for every distribution.

What does the cumulative distribution function give, and why is it the primary object?

Pick any value on the rupee scale. The distribution function at that value answers one question and nothing else: what is the probability that the standard process, observed at the horizon, finishes at or below it? Feed it Rs 90/- and it answers 0.204174. Feed it Rs 120/- and it answers 0.729601. The distribution function is a machine that turns a value into a probability, and it never returns anything outside zero and one.

Three properties come free with that definition, and they are worth naming because they are what make the object dependable. Nothing has happened far to the left, so the function reads zero there. By the far right everything has happened, so it reads one. Widening the set of outcomes being counted cannot subtract from what was already counted, so in between the function never falls.

Think of the odometer in a car. An odometer only climbs. It cannot run backwards, it starts at nothing, and if the journey is finite it settles at a final reading. Nobody has to be told any of that; it follows from what an odometer is counting. The distribution function is the odometer of a probability distribution, and its refusal to fall is not an assumption but a consequence of the fact that it is accumulating something that cannot be negative.

Now the part that decides which of the two objects is primary. The definition above never once asked whether the distribution was smooth. The definition asked only for a probability of a set, and a measure supplies that for any measurable set by construction. So a distribution function exists for every distribution, without exception: for a continuous one, for a discrete one, for a distribution that is part continuous and part discrete, and for one with a jump in the middle of it. The density has no such promise, and exactly where it runs out is set out below.

What is a density, and why is a reading of 0.019069 not a chance of 1.9 per cent?

The density is the steepness of that climb. Where the distribution function rises quickly, probability is piling up quickly, and the density is large. Where it flattens, the density is small. The definition stops there. The density is the derivative of the distribution function, wherever the distribution function has one.

Which brings the units. The distribution function is a pure number. The distribution function climbs along a scale measured in rupees. So its slope is measured in probability per rupee, and the phrase "per rupee" is not decoration on the number, it is the reason the number is not a probability. The density of the standard process at Rs 100/- is 0.019069 per rupee. Nothing on its own has a 1.9 per cent chance of anything.

The speedometer makes this concrete faster than any argument. A needle sitting on sixty is not sixty kilometres. The reading is sixty kilometres per hour, and it becomes a distance only after a duration multiplies it. Half an hour at that reading is thirty kilometres. Ask instead how far the car travels at the precise instant the needle passes sixty. An instant has no duration to multiply by, so the answer is zero. A density behaves in exactly the same way, with rupees in the place of hours.

Work it once. The density of the standard process at Rs 100/- is 0.019069 per rupee. The probability of finishing within one rupee either side of Rs 100/- is roughly that rate multiplied by the width of Rs 2/-. The product is 0.038138. The exact answer, taken as a difference of two distribution function readings, is 0.038123. The two agree to four decimal places, and the small gap is not an error in the arithmetic: it is the density leaning slightly across those two rupees rather than sitting flat. Narrow the width and the gap shrinks toward nothing. Shrinking toward nothing as the width shrinks is the signature of a rate.

How do the two connect, in each direction?

Going one way the operation is differentiation: the density is the slope of the distribution function. Going the other way the operation is to integrateAdd up a rate across a range of values, which is what converts a density into a probability.. Adding the density up from the bottom of the range to a value returns the distribution function at that value. Odometer and speedometer again. Differentiating the odometer gives the speedometer; adding the speedometer up over the trip gives the odometer.

The symmetry between the two directions hides a practical asymmetry, and the asymmetry is why working analysts reach for one of the two far more often than the other. Any question about a range answers itself with a subtraction. The probability of the standard process finishing between Rs 90/- and Rs 100/- is 0.382089 less 0.204174. The difference is 0.177915. No integral was evaluated, no width was chosen, no approximation crept in. Every range probability is a difference of two distribution function readings, and that is the reason the distribution function is the object that actually gets used.

Why does the standard process have the shape it has at the horizon?

The logarithm of the standard process at the horizon is normally distributed, with a mean of 4.665170 and a standard deviation of 0.20. The process itself is therefore the exponential of a normal quantity, and exponentiating does two things to the familiar bell. An exponential is never negative, so exponentiating cuts everything below zero away and the supportThe range of values a quantity can actually take, outside which its density is zero. is the positive numbers and nothing else. A fixed step in the logarithm becomes a larger step in rupees the higher the starting point already is, so exponentiating also stretches the upper half more than the lower half.

The result is a shape with a floor on the left and a long thin tail on the right. The lean carries one consequence above all: a shape that leans cannot have its peak, its halfway point and its average in the same place, and on the standard process those three sit at Rs 102.02/-, Rs 106.18/- and Rs 108.33/-. The order is fixed by the direction of the skewA shape that leans to one side, so that its peak, its halfway point and its average are three different values. rather than by the particular parameters: a shape leaning right always carries them in that order.

Look at the divisor in that density. Dividing by the value is what converts a rate per unit of logarithm into a rate per rupee, and it is the arithmetic reason the peak of the density does not sit where the peak of the underlying normal shape sits. The peak of the density is the modeThe value at which a density is highest, so the most likely small neighbourhood to land in., and it lands below both of the other two centres.

Why the average sits above the middle finish rather than on it is covered separately. The third centre completes the picture all three sit on. Only a density can find the most likely finish, Rs 102.02/-, because the place where a rate is highest cannot be read off a running total by inspection.

What is a state price density, and how does it differ from a probability density?

Everything so far has described what is likely. A second object on the same scale describes what something is worth, and it is built in two moves from the first. Arrow and Debreu attached their names to the idea of pricing a claim that pays if and only if one particular state comes about, and a state price density is the continuous version of that idea.

Move one: the pricing rule does not use the drift. The pricing rule grows the process at the risk-free rate of 5 per cent rather than at the 8 per cent drift, and the whole shape shifts to the left. The shifted density peaks at Rs 99.00/- rather than at Rs 102.02/-. Why the pricing rule does that, and how the second measure is built, is covered separately; here it arrives as a given object. Move two: multiply the shifted density by the discount factorThe price today of one rupee received for certain at a stated future time., 0.951229, at every value on the scale.

The result has different units, and the units are the whole of the meaning. Where a probability density reads in probability per rupee, a state price density reads in rupees today per rupee of width. At Rs 100/- it reads 0.018762. The reading says a claim paying one rupee if the process finishes in a narrow band of width delta around Rs 100/- is worth about 0.018762 times delta rupees today, and it says nothing whatever about how likely that band is.

The multiplication does less to the shape than it looks. The discount factor is one fixed number. The factor does not depend on position along the scale, so it cannot bend the shape, tilt it, or move its peak. Every height is scaled by exactly the same proportion. Whatever the shape looked like before, it looks the same afterwards, only lower everywhere by the same 4.8771 per cent.

What does each of the two total, and why is that the whole distinction?

A probability density adds up to exactly one across its whole range, and there is nothing deep about why: something has to happen, and one is what certainty is worth on the probability scale. Any function that adds up to something other than one is not describing probabilities.

A state price density adds up to 0.951229. The total of 0.951229 is worth a moment before it is accepted. Adding up the price of a rupee paid in each possible finishing state gives the price of a bundle that pays a rupee in every finishing state, and a bundle that pays a rupee whatever happens is simply a rupee received for certain at the horizon. Its price today is the discount factor. So the two objects have identical shapes to within a shift and a scale, and the only thing that separates a description of what is likely from a set of prices is what they add up to.

Which of the two is the right object to reach for, and when?

The question decides, and there are only three cases worth holding in mind. If the question names a threshold or a range, reach for the distribution function and subtract. One evaluation or two settles it, and no approximation enters. If the question is about the shape itself, where it peaks, which way it leans, whether it has more than one bump, reach for the density. A running total hides all of that in its steepness.

The third case is the one that settles the argument about which object is primary. Suppose the quantity can land on one particular value with positive probability all by itself. A quantity of that kind carries an atomA single value that carries positive probability all on its own, so a step appears in the running total there. at that value, and the distribution function handles it without complaint: it simply steps up by that amount at that point and carries on. The slope of a step is not a finite number, so the density cannot describe an atom at all. One of the two objects works in every case and the other does not, and that asymmetry is the entire reason the distribution function is treated as primary and the density as derived.

How does anybody actually work with these two objects?

The object met first in practice is the distribution function. A risk figure quoted as a percentileThe value below which a stated fraction of the outcomes falls, found by reading a distribution function backwards. is a distribution function read backwards: instead of feeding it a value and receiving a probability, the probability is fixed and the question is which value produces it. Nothing about that operation touches a density. The same is true of every threshold question anyone asks of a model, and on the standard process the answer to how much of the distribution sits at or below Rs 90/- is one reading, 0.204174, arrived at without an integral.

A researcher building the shape, rather than reading it, works the other way. Asking whether a model produces one peak or two, whether it leans left or right, whether the tail it produces is heavier than the tail it was meant to reproduce, is asking about steepness, and steepness is what a density reports. A running total that rises smoothly from zero to one looks much the same whether the shape beneath it has one bump or three.

The state price density earns its keep somewhere else again. Once it has been built for a horizon, any claim settling at that horizon can be valued in a single pass across it. The price of the claim is the payoff at each value weighted by the state price density there. Building the object once and reusing it for many claims is the practical reason it is stored as a density at all rather than recomputed for each contract, and it is entirely a model output rather than anything read off a market.

And now the everyday version. A rain gauge holds a running total, in millimetres, that climbs through a storm and never falls. The forecast talks in millimetres per hour. A rate says nothing until a duration multiplies it. Somebody asking whether the storm will overflow a drain wants the running total. Somebody asking when the storm was worst wants the rate. Both descriptions are of the same rain, and neither one can be swapped for the other without changing the question.

The standard process and its four parameters are invented.
Educational material. Not advice on any investment, tax, budget or market position.