In the EPA (1995) risk assessment, the risk posed by exposure to radon by each pathway (that is, ingestion, inhalation of radon, and inhalation of progeny) is calculated by multiplying a series of terms that describe the link between radon concentration in water and the risk to the population using that water. The terms in this link include the radon concentration in water, human exposure levels per unit concentration, radiation dose per unit exposure, and cancer risk per unit radiation dose. On the basis of the best available data, EPA developed for each of these terms probability distributions that account for the value range of a parameter and the likelihood of exceeding any value within that range. Both uncertainty and variability are accounted for in these distributions. Monte Carlo simulations were used to propagate variance in the product of the terms.

In the EPA (1995) analysis, a cancer death risk was calculated for ingestion of radon gas in drinking water with a Monte Carlo analysis and the following formula:

where

*R* = risk of fatal cancer (per person per year) associated with ingestion of radon gas in water.

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F EPA Approach to Analyzing Uncertainty and Variability
In the EPA (1995) risk assessment, the risk posed by exposure to radon by each pathway (that is, ingestion, inhalation of radon, and inhalation of progeny) is calculated by multiplying a series of terms that describe the link between radon concentration in water and the risk to the population using that water. The terms in this link include the radon concentration in water, human exposure levels per unit concentration, radiation dose per unit exposure, and cancer risk per unit radiation dose. On the basis of the best available data, EPA developed for each of these terms probability distributions that account for the value range of a parameter and the likelihood of exceeding any value within that range. Both uncertainty and variability are accounted for in these distributions. Monte Carlo simulations were used to propagate variance in the product of the terms.
Uncertainty and Variability in the Risk Posed by Ingested Radon Gas
In the EPA (1995) analysis, a cancer death risk was calculated for ingestion of radon gas in drinking water with a Monte Carlo analysis and the following formula:
where
R = risk of fatal cancer (per person per year) associated with ingestion of radon gas in water.

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C = concentration of radon in water, pCi/L.
F = fraction of radon remaining in water at time of ingestion.
V = volume of water ingested, L/d.
RF = ingestion risk factor for radon gas (cancer-death risk per person per picocurie ingested).
Table F.1 summarizes the probability distributions used to represent value ranges for the parameters of this model. Each parameter has a variability distribution that is defined by two parameters, such as the mean and standard deviation. The two parameters are drawn from two distributions that represent their uncertainty. The concentration of radon in water, C, was derived by EPA from the National Inorganics and Radionuclides survey (Longtin 1990). A population-weighted population density function (PDF) was developed that was fitted to a lognormal distribution having a geometric mean of 200 pCi/L and a GSD of 1.85. The uncertainty in this distribution was obtained by using the Student t-distribution and the inverse chi-squared distribution to simulate the resampling of mean values and standard deviations. A sample size of 10 was used to resample from the Student t and inverse chi-squared distributions. Even though some 1,000 water systems are represented in the survey set, the assumed small sample size was selected to reflect the fact that the population-weighted concentration distribution was dominated by a small number of large water supplies for which there were a limited number of measurements. The variability of F, the fraction of radon remaining in water, was modeled as a beta distribution in the interval 0.1–0.3 with an uncertain mean and mode. The uncertainty of the mean was modeled as a uniform distribution in the range 0.7-0.9, and the uncertainty in the mode was modeled as a uniform distribution between 0.5 and the sampled mean value.
Uncertainty and Variability in Risk Posed by Inhaled Radon Gas
In the EPA uncertainty analysis, the basic equation used to calculate inhalation uptake of radon gas is based on both the uncertainty and the variability of the unit dose factor, but the risk factor per unit dose is based on a single value. The unit dose factor for gas released from water includes three factors:
where
UD = unit dose (pCi inhaled per year per pCi/L of radon in water).
TF = transfer factor, which is the increase in radon concentration in indoor air per unit radon concentration in water (pCi/L[air] per pCi/L[water]).
BR = breathing rate (L/d).
OF = occupancy factor (fraction of time person spends indoors).

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Table F.1
Probability Density Functions Used in the Calculation of Risk of Cancer Posed by Ingestion of Radon Gas in Water (following notation of EPA 1995)
Variability
Uncertainty
Parameter
Distribution Type
Distribution Values
Distribution Type
Distribution Values
C (concentration of radon in water, pCi/L[water])
Lognormal distribution, LN(µ)
C = LN(µ,σ) µ = TS(m,s,n) (σ)2=IChi(s,n)
Student t, distribution, TS(m,s,n) Inverse chi-squared distribution, IChi(s,n)
n= 10 m = ln(200) s = ln(1.85)
V (volume of water ingested, L/d)
Lognormal distribution, LN(µ,σ)
V = LN(µ,σ) µ=TS(m,s,n) (σ)2=IChi(s,n)
Student t distribution, TS(m,s,n) Inverse chi-squared distribution, IChi(s,n)
n=100 m = ln(0.526) s = ln(1.922)
F (fraction remaining)
Beta distribution, B(m,md,min,max)
F=B(m,md,min,max) min=0.5 max=1 m=U(a,b) md=U(m,max) or U(min,m)
Uniform distributions U(a,b) U(m,max)
a=0.7 b=0.9 min = 0.5 max=1
RF (risk factor, cancer-death risk per person per pCi ingested)
This factor has uncertainty only
RF=LN(µ,σ)
µ= ln(1.24 × 10-11 σ = ln(2.42)
Calculated individual risk
Uncertainty==> Variability 5th percentile mean 95th percentile
5th percentile 1.7 × 10-8 1.3 × 10-7 4.0 × 10-7
median 8.3 × 10-8 6.2 × 10-7 1.9 × 10-6
95th percentile 3.4 × 10-7 2.6 × 10-6 7.9 × 10-6
m=mean value derived from a sample
s=standard deviation of a sample
n=sample size
md=mode of a sample
µ=mean value of ln(x) in a lognormal distribution
µ=standard deviation of ln(x) in a lognormal distribution

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Table F.2 summarizes the probability distributions used to represent value ranges for the parameters of this model. Each parameter has a variability distribution that is defined by two parameters, such as the mean and standard deviation. They are drawn from distributions that represent the uncertainty of the two parameters. EPA used two-dimensional Monte Carlo simulations to develop an outcome distribution for UD. Two models were used to develop a distribution for the transfer factor (TF)—a one compartment indoor-air model and a three-compartment indoor-air model. Similar results were obtained from the two models. Because of the lack of information on the uncertainty and variability in the inhalation risk factor for radon gas, EPA calculated the mean population risk, PR, as
where
PR = population risk of fatal cancer (cancers per year) posed by ingestion of radon gas in water.
UD = unit dose (pCi inhaled per year per pCi/L of radon in water).
RF = risk factor, lifetime risk of cancer per person per pCi inhaled per year.
Cmean = population mean concentration of radon in water, pCi/L.
N number of people in the population.
EPA used an inhalation risk factor, RF, of 1.1 × 10-12 cancer death per person per pCi/L of radon in water and a Cmean, of 246 pCi/L. We can use this equation with N = 1 and UD = 380, which is the median value with respect to uncertainty and the mean value with respect to variability, to estimate the per caput risk within the exposed population. This gives a per caput risk of 1.0 × 10-7, which is low compared with the mean individual risk (at median with respect to uncertainty) of 6.2 × 10-7 calculated for the ingestion pathway.
Uncertainty and Variability in Risk Posed By Inhaled Radon Progeny
The EPA (1995) report provided a separate calculation of variability and uncertainty of risk associated with exposure to radon progeny attributable to radon releases from household uses of water. In the EPA uncertainty analysis for radon progeny, the basic equation used to calculate risk is based on uncertainty and variability of the unit dose factor. The unit dose factor for radon progeny released from water includes three factors:
where
UD= unit dose (WLM per year pCi/L)

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Table F.2
Probability Density Functions Used by EPA (1995) in Calculation of Unit Dose of Radon Gas Inhaled after Transfer from Water (following notation of EPA 1995)
Variability
Uncertainty
Parameter
Distribution Type
Distribution Values
Distribution Type
Distribution Values
TF (transfer factor, pCi/L[air] per pCi/L[water])
Truncated lognormal TLN(µ, σ,min,max)
TF = TLN(µ, σ, min,max) µ = TS(m,s,n) (σ=2=IChi(s,n) min=6. × 10-6 max=8. × 10-4
Student t distribution, TS(m,s,n) Inverse chi-squared distribution, IChi(s,n)
n= 25 m = ln(6.57 × 10-5) s = ln(2.88)
BR (breathing rate, L/day)
Truncated normal TN(m,s,min, max)
BR = TN(µσ min.max) µ = TS(m,s,n) σ2=IChi(s,n) min=3700 max=66,000
(Student t distribution, TS(m,s,n) Inverse chi-squared distribution, IChi(s,n)
n=10 m = 13,000 s = 2880
OF (occupancy factor)
Beta distribution B(m,md,min,max)
OF=B(m,md,min,max) Min=0.17 Max=0.95 m=U(a,b) md=U(min,m) U(m,max)
Uniform distributions U(a,b) U(min,m) U(m,max)
a=0.65 b= 0.80 min = 0.17 max = 0.95
UD pCi inhaled per(unit dose, year per pCi/L)
Uncertainty==> Variability 5th percentile 17 mean 95th percentile
5th percentile 32 250 800
median 57 380 1300
95th percentile 540 2400
m=mean value derived from a sample
s=standard deviation of a sample
n=sample size
md=mode of a sample
µ=mean value of In(x) in a lognormal distribution
σ=standard deviation of ln(x) in a lognormal distribution
µ=mean value of x in a normal distribution
µ=standard deviation of x in a normal distribution

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TF = transfer factor, which is the increase in radon concentration in indoor air per unit radon concentration in water [pCi/L(air) per pCi/ L(water)].
EF = equilibrium factor (fraction of potential alpha energy of radon progeny that actually exists in indoor air compared with the maximum possible alpha energy under true equilibrium conditions).
OF = occupancy factor (fraction of time that person spends indoors).
From the UD, EPA calculated the unit risk factor, UR (lifetime risk of cancer death per pCi/L), and IR, the individual risk of cancer per individual:
where
RF = risk factor, lifetime risk of cancer death per person-WLM of exposure.
C = concentration of radon in water, pCi/L.
Table F.3 summarizes the probability distributions used to represent value ranges for the parameters of this model. Each parameter has a variability distribution that is defined by two parameters, such as the mean and standard deviation. They are drawn from other distributions that represent the uncertainty. EPA again used two-dimensional Monte Carlo simulations to develop an outcome distribution for UD. As for the radon-gas inhalation model, both a one-compartment and a three-compartment model of indoor air were used, and they produced similar results for the transfer factor.
Table F.4 summarizes the calculation of PDFs used by EPA (1995) in the calculation of unit dose of radon progeny inhaled after transfer from water. From this table, we observe that the lifetime mean individual risk (at the median with respect to uncertainty) is 1.3 × 10-6 per year of exposure for the inhalation of radon progeny from water use. That is about twice the risk calculated for the ingestion pathway, 6.2 × 10-7, and 10 times the mean (median) risk for inhalation of radon gas, 1.0 × 10-7.

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Table F.3
Probability Density Functions Used by EPA (1995) in Calculation of Unit Dose of Radon Progeny Inhaled after Transfer from Water (following notation of EPA 1995)
Variability
Uncertainty
Parameter
Distribution Type
Distribution Values
Distribution Type
Distribution Values
TF (transfer factor, pCi/L[air] per pCi/L[water])
Truncated lognormal TLN(µ,σ,min,max)
TF = TLN(µ,σ,min,max)CARRIAGE>(µ) = TS(m,s,n) (σ)2=IChi(s,n) min=6. × 10-6 max=8. × 10-4
Student t distribution, TS(m,s,n) Inverse chi-squared distribution, IChi(s,n)
n= 25 m = ln(6.57 × 10-5) s = ln(2.88)
EF (equilibrium factor)
Beta distribution B(m,md,min,max)
EF=B(m,md,min,max) min=0.1 max=0.9 m=U(a,b) md=U(min,m) md=U(m,max)
Uniform distributions U(a,b) U(min,max) U(m,max)
a=0.35 b= 0.55 min = 0.1 max=.09
OF (occupancy factor)
Beta distribution B(m,md,min,max)
OF=B(m,md,min,max) min=0.17 max=0.95 m=U(a,b) md=U(min,m) md=U(m,max)
Uniform distributions U(a,b) U(min,m) U(m,max)
a=0.65 b= 0.55 min = 0.1 max=.09
UD (unit dose, WLM/y per pCi/L)
Uncertainty==> Variability 5th percentile mean 95th percentile
5th percentile 6.5 × 10-7 1.2 × 10-5 3.9 × 10-5
median 1.2 × 10-6 1.8 × 10-5 6.4 × 10-5
95th percentile 2.1 × 10-6 2.7 × 10-5 1.0 × 10-4
m=mean value derived from a sample
s=standard deviation of a sample
n=sample size
md=mode of a sample
µ=mean value of In(x) in a lognormal distribution,
µ=standard deviation of In(x) in a lognormal distribution

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Table F.4
Probability Density Functions Developed by EPA (1995) in Calculation of Risk, Unit Risk and Variation of Individual Risk Posed by Radon Progeny Inhaled after Transfer from Water (following notation of EPA 1995)
RF (risk factor, lifetime risk of cancer death per person-WLM)
This factor has uncertainty only
RF=LN(µ,σ
µ= ln(2.83 × 10-4) σ = ln(1.53)
UR (unit risk factor, lifetime risk of cancer death per pCi/L)
Uncertainty = => Variability 5th percentile mean 95th percentile
5th percentile 1.3 × 10-10 2.1 × 10-9 7.2 × 10-8
median 3.4 × l0-10 5.1 × l0-9 1.8 × l0-8
95th percentile 8.9 × 10-10 1.2 × 10-8 4.2 × 10-8
C (concentration of radon in the water, pCi/L [water])
Lognormal distribution LN(µ,σ)
C = LN(µl,σl,) µ = TS(m,s,n) (σ)2=IChi(s,n)
Student t distribution, TS(m,s,n) Inverse chi-squared distribution, IChi(s,n)
n= 10 m = ln(200) s = ln(1.85)
IR (individual risk, lifetime risk of cancers per person-year)
Uncertainty==> Variability 5th percentile mean 95th percentile
5th percentile 1.8 × 10-8 5.2 × 10-7 1.9 × 10-6
median 5.4 × 10-8 1.3 × 10-6 5.0 × 10-6
95th percentile 1.5 × 10-7 3.2 × 10-6 1.3 × 10-5

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