Flashcards
VI. Bernoulli trials
Cards in this deck (7)
- 1Bernoulli trialsRepeated independent trials are called Bernoulli trials if there are only two possible outcomes for each trial and their probabilities remain the same throughout the trials.
- 2Binomial distributionLet b(k; n,p) be the probability that n Bernoulli trials with probabilities p for success and q=1-p for failure result in k siccesses and n-k failures. Then
b(k; n,p) = C(k,n) * p^k * q^(n-k) - 3The law of large numbersAs the number of trials increases, the probability that the average number of successes deviates from the probability of success by more than any preassigned e tends to zero.
- 4The Poisson approximation to the binomial distributionb(k; n,p) ~ e^-L * L^k / k!
where L = n*p - 5The Poisson distributionConsider a sequence of random events occuring in time. Each event is represented by a point on the time axis. Then p(k,L) is a chance of having exactly k events on a unit interval. For a time interval of length t:
p(k; L*t) = e^(-L*t) * (L*t)^k / k! - 6The negative binomial (Pascal) distributionConsider a succession of Bernoulli trials. Then the probability that r-th success occurs at the trial number r+k (where k = 0,1,...) is:
f(k; r,p) = C(k,r+k-1) * p^r * q^k - 7The multinomial distributionConsider n repeated independent trials, where each trial can have one of several outcomes: E1, ..., Er. Suppose that pi is the probability of realization of Ei in each trial. Then the probability that in n trials E1 occurs k1 times, E2 occurs k2 times etc. is:
p1^k1 * p2^k2 *... * pr^kr * n! / (k1! * k2! * ... * kr!)
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- Probability Theory and It's Applications - W. Feller Flashcards on VI. Bernoulli trials, created by Sergei Fomin on 22/08/2017.
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