NE7303 Handout-16 Counting Statistics PDF

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COUNTING STATISTICS OBJECTIVES: - UNDERSTAND RANDOMNESS OF DECAY PROCESS - STATISTICAL MODELS FOR RANDOM TRIALS - CALCULATE STANDARD DEVIATIONS ON MEASUREMENTS - LOWER LIMIT OF DETECTION STATISTICAL MODELS FOR RANDOM TRIALS 1) Binomial

2) Poisson, 3) Normal or Gaussian Distribution

1) BINOMIAL PROBABILITY DISTRIBUTION: - Binomial process- Trial can have only two outcomes - For processes occurring with constant probability e.g., flipping a coin

• P(x) is the predicted probability distribution function, as given by the binomial distribution and is defined only for integer values of N and x. • N is total number of trials • p is probability of success • x is number of successes

2) POISSON PROBABILITY DISTRIBUTION: - Simplification of binomial distribution with certain constraints - Applies if success probability p is small and constant, and the trials occur independently of each other. - Example: Radioactive decay and detection are Poisson random processes

This has the same properties as Binomial distribution. (a) It is normalized

(b) Mean, variance, and standard deviation are given by:

For values of mean > 25 the Poisson distribution is very well approximated by the Gaussian (or “normal”) distribution for which certain confidence levels have been established in terms of the standard deviation. 3) GAUSSIAN OR NORMAL DISTRIBUTION: - The binomial and Poisson distributions become similar and tend to approach the shape of a NORMAL, or GAUSSIAN, distribution as p gets small (P...