# What is binomial continuous distribution

## Binomial distribution Poisson distribution (borderline case of the binomial distribution) Continuous probability distribution

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1 Basics of Statistics Probability distributions Discrete probability distributions Binomial distribution Poisson distribution (limit case of binomial distribution) Continuous probability distribution Standard normal distribution (limit case of Poisson distribution) 1

2 Basics of Statistics Binomial Distribution Example: Counting measurement of α-particles During the measurement time n particles are emitted; Every particle is detected with the same probability p; Expected value of the detected particles: Variance of the detected particles: np np (1-p) 2

3 Basics of statistics Binomial distribution Discrete probability distribution for an alternative feature (e.g. success or failure) Experiment from n independent repetitions with the same probability of success p Expected value Variance np np (1-p) P n, p () n !! (nnp (1 p)) ! 3

4 Pn, p () Basics of statistics Binomial distribution n 10 Expected value 3 p 0.3 Variance 2.1 Pn, p () 0 0,,,,,,,,,,, 9049E-06 0.3 0.25 0 , 2 0.15 0.1 0.05 0 binomial distribution

5 Basics of Statistics Poisson Distribution Borderline case of the binomial distribution for very large n: Very large number of event possibilities with the same (very small) event probability Expected value λ (generally λ <20) Variance λ P () e! 5

6 P () Fundamentals of statistics Poisson distribution Expected value lambda 5 Variance lambda 5 P () 0 0,,,,,,,,,,,,,,,,, 9139E-05 0.2 0.15 0.1 0.05 0 Poisson distribution

7 Basic principles of statistics Standard normal distribution Continuous probability distribution: Expected value μ variance σ 2 y () 1 () 2 ep Limit case of the Poisson distribution for large expected values ​​and for variance σ 2 = μ 7

8 y () Basic principles of statistics Standard normal distribution Expected value µ 100 Standard normal distribution 3 sigma = 3 sigma = 10 sigma =. 0645E-243 7.69471E-24 5.14101E, 14 1 4.4715E-238 2.08121E-23 5.74198E, 5326E-233 0.125.57308E-23 6.40609E, 2836E-228 1.47752E-22 7.13907E, 8213E-2240.1 3.87817E-22 7.94708E, 3625E-219 1.00781E-21 8.83672E, 5795E-215 0.082.5929E-21 9.81504E, 788E-210 6.60468E-21 0,, 1073E-206 0.061.66561E-20 0,, 1096E-201 4.15866E-20 0,, 9122E-197 0.041, 02799E-19 0,, 0235E-192 2.51584E-19 0,, 9084E-188 0.026.09585E-19 0,, 1839E-184 1.46232E-18 0,, 7536E, 47301E-18 0,, 3506E-176 8.16636E-18 0,, 592E-172 1.90111E-17 0, sigma = 3 sigma = 10 sigma =

9 P () y () Fundamentals of statistics distributions Expected value µ 100 sigma 10 y () 0 7.69471E, 08121E, 57308E, 47752E, 87817E, 00781E, 5929E, 60468E, 66561E, 15866E, 02799E, 51584E, 09585E, 46232E, 47301E, 16636E, 90111E-17 0.05 0.04 0.03 0.02 0.01 0 Standard normal distribution Expected value lambda 100 Variance lambda 100 P () 0 3.7201E, 7201E, 86E, 2001E, 55E, 1001E, 1668E, 3811E, 2264E, 0252E, 0252E, 3196E, 7663E, 9741E, 2672E, 8448E, 778E-25 0.05 0.04 0.03 0.02 0.01 0 Poisson distribution

10 y () Applications of Statistics Detection Limit sigma =, 69471E, 08121E, 57308E-23 0.05 3 1.47752E, 87817E-22 0.04 5 1.00781E, 5929E-21 0.03 7 6.60468E, 66561E, 15866E-20 0,, 02799E, 51584E-19 0,, 09585E, 46232E, 47301E, 16636E, 90111E-17 Expected value µ 100 sigma Standard normal distribution Detection limit e.g. µ + 1.645 sigma quantile e.g. 5% 10

11 y () Applications of statistics Error of the first kind false positive decision sigma =, 69471E, 08121E, 57308E-23 0.05 3 1.47752E, 87817E-22 0.04 5 1.00781E, 5929E-21 0.03 7 6 , 60468E, 66561E, 15866E-20 0,, 02799E, 51584E-19 0,, 09585E, 46232E, 47301E, 16636E, 90111E-17 Expected value µ 100 sigma e.g. Decision that an activity is present in a sample, although the sample is activity-free or decision that a person has an occupational Cs-137 activity, although only the Cs-137 activity due to civilization is present, or a wrong decision that one People have a certain disease. Standard normal distribution. Detection limit eg µ + 1.645 sigma quantile (type 1 error) e.g. 5% 11

12 y () Applications of statistics Detection limit 0.045 Expected value µ 100 Background effect Measurement effect 10 0.04 0 7.69471E-24 1.76799E, 035 2.08121E-23 6.64487E, 57308E-23 0.03 2.47257E, 47752E- 22 9.10897E, 025 3.87817E-22 3.32236E-38 quantile 5 1.00781E-21 0.02 1.9972E, 5929E-21 (error 4.28915E type) 7 0.015 6.60468E-21 1.51817E -36 e.g. 5% 8 1.66561E-20 0.01 5.32017E, 15866E-20 1.84581E-35 0,, 02799E-19 6.34026E, 51584E, 15617E, 09585E-19 7.25967E, 46232E-18 2.41996E , 47301E-18 7.98647E, 16636E-18 2.60951E, 90111E-17 8.44153E-32 Detection limit standard normal distribution e.g. µ + 2 * 1.645 sigma background measurement effect 12

13 y () Applications of statistics Error of the second kind False negative decision e.g. Decision that there is no activity in a sample, although the sample contains an activity or decision that in an expected value humans only the µ civilizationally related 100 Cs-137 activity is present although an occupationally related Cs-137 incorporation standard normal distribution is present, e.g. µ + 2 * 1.645 sigma wrong decision that a person does not have a certain disease 0, background measurement effect background measurement effect 10 0.04 0 7.69471E-24 1.76799E, 035 2.08121E-23 6.64487E, 57308E-23 0.03 2.47257E, 47752E-22 9.10897E, 025 3.87817E-22 3.32236E-38 quantile 5 1.00781E-21 0.02 1. 19972E, 5929E-21 (error 4.28915E type) 7 0.015 6.60468E-21 1.51817E-36 e.g. 5% 8 1.66561E-20 0.01 5.32017E, 15866E-20 1.84581E-35 0,, 02799E-19 6.34026E, 51584E, 15617E, 09585E-19 7.25967E, 46232E-18 2.41996E , 47301E-18 7.98647E, 16636E-18 2.60951E, 90111E-17 8.44153E-32 detection limit

14 Applications of statistics Errors in dosimetry Error components Measurement of body or organ activity Counting statistics Detector or patient positioning Individual variation of the body dimensions Individual variation of the activity distribution in the body Calibration error Error in the spectrum evaluation Calculation of the dose from the body or organ activity Individual variation of the body dimensions Individual variation of biokinetic behavior 14

15 Applications of statistics Errors in dosimetry Source of error (Type A: purely statistical Type B: not purely statistical) Low energy E <20 kev Average spread in% Average energy 20 kev 100 kev Counting statistics (A. ) Detector positioning (B) 20 5 <5 Background (B) <5 Body dimensions (B) Absorbent structures (B) Activity distribution (B) 30 5 <5 Calibration (B) Spectrum evaluation (B)

16 Applications of statistics Error propagation Example: Measurement of activity in the body Counting statistics σ (z) Detector or patient positioning σ (pp) Individual variation of the body dimensions σ (kd) Individual variation of the activity distribution in the body σ (av) Calibration error σ (kf) Error in the spectrum evaluation σ (sa) A f (Z, PP, KD, AV, KF, SA) (A) f Z 2 2 (Z) f PP 2 2 (PP) ... f SA 2 2 (SA) 16

17 Effects of statistics interlaboratory comparisons Example: measurement of Cs-137 activity in a phantom (interlaboratory comparison of the Federal Office for Radiation Protection) 17

18 Effects of statistics interlaboratory comparisons Example: Interpretation of measurement data for I-131 activity in the thyroid (interlaboratory comparison by IAEA / IDEAS) E (50) - Ratio to GM Effective subsequent dose in msv Minimum 1.58E-05 Maximum 199 GM 2.45 GSD 1.37 ID 18