Learning about a Binomial Probability

2019 ◽  
pp. 217-265
Author(s):  
Jim Albert ◽  
Jingchen Hu
Keyword(s):  
Binomial Probability

SOME TESTS OF BINOMIAL PROBABILITY ASSESSMENTS

1978 ◽  
Vol 9 (4) ◽  
pp. 564-576
Author(s):  
William S. Peters
Keyword(s):  
Binomial Probability

Mitotic Index of Invasive Breast Carcinoma

2009 ◽  
Vol 133 (11) ◽  
pp. 1826-1833 ◽  
Author(s):  
John S. Meyer ◽  
Eric Cosatto ◽  
Hans Peter Graf
Keyword(s):  
Breast Carcinoma ◽  
Mitotic Index ◽  
Clinical Decision Making ◽  
Clinical Decision ◽  
Mitotic Cell ◽  
Standard Errors ◽  
Mitotic Figure ◽  
Binomial Probability ◽  
Breast Carcinomas ◽  
Coefficients Of Variation

Abstract Context.—Mitotic figure counts are related to breast cancer behavior but have not been sufficiently reproducible to be accepted for clinical decision-making. Objective.—To improve reproducibility and accuracy of the mitotic count. Design.—Mitotic index (MI) was defined as the mitotic cell count per 10 high-power fields (HPFs), an area 0.183 mm2. Two to 6 replicate sets of 10 HPFs were counted from 328 invasive breast carcinomas. Standard errors and coefficients of variation for mean MI were compared with expected results predicted by the binomial distribution. Results.—The boundaries for MI that separated the data into equal thirds (tertials) were 1.14 and 5.33. Standard errors and coefficients of variation for MI followed distributions predicted by binomial probability. Mean coefficient of variation was 147% for the low tertial, 72% for the midtertial, and 34.6% for the upper tertial. Conclusions.—Standard errors for MI based on a single count of 10 HPFs are too broad and coefficients of variation too large to be acceptable for clinical use. This is explained as a binomial probability effect, possibly with a contribution from tumor heterogeneity. Errors can be reduced in proportion to the square root of the number of sets of 10 HPFs counted. Tertial cutoffs of MI of the Nottingham system currently used in breast carcinoma grading are too high to be applicable to the population we studied. We recommend validation of cutoffs before they are applied to a particular population of breast carcinomas. Counting 5 sets of 10 HPFs is necessary to accurately rank carcinomas with low MIs.

On the Distribution of Sum of Independent Positive Binomial Variables

1970 ◽  
Vol 13 (1) ◽  
pp. 151-152 ◽  
Author(s):  
J. C. Ahuja
Keyword(s):  
Power Series ◽  
Inversion Formula ◽  
Probability Function ◽  
Random Variables ◽  
Random Variable ◽  
Characteristic Functions ◽  
Binomial Probability ◽  
Generalized Power Series ◽  
Alternative Derivation ◽  
Image Position

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.

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