distributions

Question 1

binomial distribution

Answer 1

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

Question 2

geometric distribution

Answer 2

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

(this is a kind of discrete exponential distribution)

Question 3

negative binomial

Answer 3

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

Question 4

hypergeometric

Answer 4

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)

Question 5

Poisson distribution

Answer 5

“chained” iid exponential distributions, in a fixed space or time interval s; random variable X = number of events in s

Question 6

multinomial distribution

Answer 6

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

Question 7

uniform distribution

Question 8

exponential distribution

Answer 8

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)

Question 9

gamma distribution

Answer 9

“chained” iid exponential distributions; assuming r events, random variable X = length of time (or distance) for the rth event to occur

Question 10

Weibull distribution

Answer 10

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)

Question 11

beta distribution

Answer 11

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}

Question 12

normal distribution

Answer 12

• 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

Question 13

lognormal distribution

Answer 13

if log(X) is normally distributed, then X has a lognormal distribution

Question 14

Chi-squared distribution

Answer 14

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

Question 15

Student’s t-distribution

Answer 15

• for sample mean distribution, when population variance is unknown
  
  • 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
  
  • the denominator is the sample mean standard error estimate
  • if using the CLT, the denominator would reamin 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

Question 16

F-distribution

Answer 16

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

Question 17

central limit theorem

Answer 17

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

Question 18

Poisson process

Answer 18

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