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 : (18)

• exponential distribution (19)

## B..2 Cumulative Distribution Functions

• standard normal : (20)

where is the error function.

• two-side truncated normal [10, pp.156-162]: (21)

where and are given by Equation 11.

• exponential : (22)

• shifted and right-truncated exponential: (23)

## B..3 Moments of a Truncated Normal

These can be found in the literature, e.g. in . 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.

### B..3.2 Variance  (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 .

### 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 .

#### 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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