Subsections

# 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 distribution [16]:

 (18)

• exponential distribution [16]

 (19)

## B..2 Cumulative Distribution Functions

• standard normal [16]:

 (20)

where is the error function.

• two-side truncated normal [10, pp.156-162]:

 (21)

where and are given by Equation 11.

• exponential [16]:

 (22)

• shifted and right-truncated exponential:

 (23)

## 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 .

### B..3.1 Expected Value

 (24)

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.

 (25)

## 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 distribution8and Equation 19

where the shift of the exponential by is already taken into account. From lists of integrals of exponential functions9

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

 (26)

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

 (27)

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

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

#### Footnotes

... distribution8
http://en.wikipedia.org/wiki/Truncated_distribution
... functions9
http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions
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