distributions Flashcards

1
Q

binomial distribution

A

fixed number, N, of Bernoulli trials, with random variable X = # of successes after the N trials

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2
Q

geometric distribution

A

infinite number of Bernoulli trials, with random variable X = # of trials until 1st success

(this is a kind of discrete exponential distribution)

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3
Q

negative binomial

A

infinite number of Bernoulli trials, with random variable X = # of trials until r successes

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4
Q

hypergeometric

A

N items, with r<=N being a certain kind; choose n<=N items without replacement; random variable X = # of items of type r being in that sample of n

(eg fish stocking experiment, distribution of tagged catch proportions)

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5
Q

Poisson distribution

A

“chained” iid exponential distributions, in a fixed space or time interval s; random variable X = number of events in s (one of the two main distributions associated with Poisson process)

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6
Q

multinomial distribution

A

fixed number, N, of multinomial trials, with probability of items 1 thru k being p1,…,pk, random variable X = a fixed event count tuple for items 1 thru k

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7
Q

uniform distribution

A
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8
Q

exponential distribution

A

continuous pdf, of type L exp(-L x), on [0,inf)

mean=1/L; var=1/L^2

has the memoryless property–P(X>=x) is again an exponential distribution with parameter L (renormalizing)

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9
Q

gamma distribution

A

“chained” iid exponential distributions; assuming r events, random variable X = length of time (or distance) for the rth event to occur (one of two main distros associated with Poisson process)

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10
Q

Weibull distribution

A

models a broad range of random variables, largely of the nature of a time to failure or time between events

a L^a x^{a-1} exp(-(Lx)^a)

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11
Q

beta distribution

A

used to model proportions

related to binomial distribution–the beta distribution has the number of successes as a parameter, and the (binomial) “probability” (or proportion of successes) as the random variable (Beta can be considered a conjugate prior to binomial, among others)

(Gam(a+b)/(Gam(a)Gam(b))) x^{a-1} (1-x)^{b-1}

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12
Q

normal distribution

A
  • X ~ N(mu,sig^2)
  • (1/(sig sqrt(2 pi))) exp(-(x-mu)^2 / (2 sig^2))
  • 68% is within 1 std dev of mean
  • 95% is within 2 std devs of mean
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13
Q

lognormal distribution

A
  • if log(X) is normally distributed, then X has a lognormal distribution
  • note mu and sigma are for the log of the RV values
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14
Q

Chi-squared distribution

A

given iid X_i ~ N(0,1), random variable Y=X_1^2+…+X_k^2 has a Chi-square distribution with k degrees of freedom

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15
Q

Student’s t-distribution

A
  • for sample mean distribution, when population variance is unknown (population assumed approximately normal)
    • if pop. variance known, then can use iid with that variance -> CLT
    • Student’s allows using sample variance in lieu
    • the distribution of sample variance is Chi-square, provided the population is approximately normal (Cochran’s theorem)
  • form the t-statistic, (x-mu) / (sig/sqrt(n)), for n samples with mean x, (unbiased) sample variance sig^2 (Bessel correction)
    • the denominator is the sample mean standard error estimate (from linear comb. of r.v.’s)
    • if using the CLT, the denominator would remain the same, with sig^2 now population variance
  • the t-statistic has t-distribution, N(0,1) / sqrt(X^2 / v) (Hayter’s formulation)
    • X^2 is a Chi-square distribution
    • v is the degrees of freedom, which equals n-1 for Student’s
    • the distributions (top and bottom) are independent
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16
Q

F-distribution

A

X_v1^2/v1 / X_v2^2/v2

each X_vi is an independent Chi-square distribution with vi degrees of freedom

comes up in eg computation of t-distribution’s pdf

17
Q

central limit theorem

A

the average of a set of n iid random variables converges to N(mu,sig^2 / n) in n

for similar theorems for linear combinations of iid, or even just independent variables, see Lindberg

18
Q

Poisson process

A

a discrete stochastic process, amounting to a series of events with identical exponential distributions

has two common distributions associated with it, the Poisson distribution, and the Gamma distribution