- B..1 Density Functions
- B..2 Cumulative Distribution Functions
- B..3 Moments of a Truncated Normal

- B..4 Moments of a Shifted Truncated Exponential

B. Formulas and Derivations

For completeness, we give here the rest of the formulas not given throughout the paper, and the derivations of those not found in the literature.

B..1 Density Functions

- standard normal [16]:

where is the error function. - two-side truncated normal [10, pp.156-162]:

where and are given by Equation 11. - exponential [16]:

- shifted and right-truncated exponential:

B..3 Moments of a Truncated Normal

These can be found in the literature, e.g. in [10]. Let be a normally-distributed random variable with mean and variance , which we left-truncate at and right-truncate at .

We do not us the sign at the upper limit of here (and in the equations below) to denote that the right-truncation is an option (i.e. can be ) in the context of this paper.

B..4 Moments of a Shifted Truncated Exponential

We have not found those in the literature. Let be an exponentially distributed random variable with rate parameter , which we shift by and right-truncate at .

B..4.1 Expected Value

From the definition of the expected value of a truncated
distribution^{8}and Equation 19

where the shift of the exponential by is already taken into account. From lists of integrals of exponential functions

Putting the last 2 equations together and working out the calculation leads to

For only shift but no truncation ( , ), and , so Equation 26 becomes

which for a zero shift ( ) it becomes , as expected [16].

B..4.2 Variance

We can break down a shifted to a mixture of its right-truncated and left-truncated parts weighted by and where . The two parts are non-correlated, so for their variances it holds that

Since shifts do not affect variances, . Moreover, , leading to

For only shift but no truncation ( , ), and Equation 27 becomes

as expected; the shift does not affect the variance [16].

- ...
distribution
^{8} `http://en.wikipedia.org/wiki/Truncated_distribution`- ...
functions
^{9} `http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions`