EXPON.DIST

What it does

EXPON.DIST(x, lambda, cumulative)

EXPON.DIST(x, lambda, cumulative) models a nonnegative waiting time at a positive rate lambda. TRUE returns the probability of a wait no longer than x; FALSE returns a density at x. Rate has inverse-time units, so if x is in hours, lambda is per hour. The mathematical mean wait is 1/lambda.

Try it

=EXPON.DIST(2,0.5,TRUE) is about 0.632120558828558: the rate-times-time product is one. About 63.2% of the modeled waits finish within two time units. Changing TRUE to FALSE gives density about 0.183939720585721.

=EXPON.DIST(0,2,FALSE) returns 2. A density can exceed one; it describes probability per unit of time. It is not the probability of a wait lasting exactly zero.

Quirks

Negative x or a nonpositive rate gives #NUM!; nonnumeric numeric arguments give #VALUE!. Zero time is admitted: the CDF is zero and the density is the rate. Do not pass a mean waiting time as lambda without taking its reciprocal.

A naive 1-exp(-lambda*x) can erase a small positive CDF when the exponential rounds to one. The current route has an explicit correction for this case. It also checks the stored rate-times-time product for overflow and flushes a subnormal product to zero, so a representable small CDF is not enough by itself to predict the published result.

How to implement

The current kernel uses R for the stored arithmetic operation, with the measured public LN and EXP leaves.

1. Validate x>=0 and lambda>0. Store t=R(lambda*x). A nonfinite t gives #NUM!; replace a subnormal t with zero.
2. For density, obtain u=EXP(-t) and return R(lambda*u) under the finite-result policy.
3. For the CDF, obtain the same u. If it is exactly one, return t.
4. Otherwise store d=R(u-1). If t<1, store c=R(R(d*(-t))/LN(u)); otherwise use c=d. Return -c. Keep the product and division stores separate.
5. The current finite-result helper turns nonfinite answers into #NUM! and flushes final subnormal answers to zero.

For the main example, t=1, u is about 0.36787944117144233, and the CDF is about 0.632120558828558. The density multiplies u by 0.5. At t=1e-20, the public exponential rounds to one; returning t preserves the small CDF that direct subtraction would lose.

This is the stored Kahan correction also used by the PMT discount recipe. The leaves and operation order are part of the candidate; substituting a language’s exp or expm1 can change final bits. The problem report retains a numerical residual, so this is not a whole-domain equality claim.

Rust code

63 expon_dist "EXPON.DIST" (3, 3) [V "x", V "lambda", V "cumulative"] |s, a, sv| {
    let [x, lambda] = nums::<2>(s, a)?;
    out(s, dd::expon_dist_kernel(x, lambda, flag(s, a, 2)?))
}

No implementation has been published with a Handbook test suite for this function.

Version differences

The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.

The record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.

Recorded environments and comparison details.

Alternatives

GAMMA.DIST with shape one is related, but its scale transformation has its own stores. WEIBULL.DIST generalises the waiting-time shape.

Named variants

No named variants are recorded yet.