Special Distributions Flashcards
Describe Bernoulli trial
Discrete distribution with 2 outcomes. X=1 is success and X=0 is failure. Special case of binomial where n=1.
E(X) & V(X) for Bernoulli
E(X) = P V(X) = P(1 - P)
What is a binomial distribution?
Repeated applications of Bernoulli trial, where each individual has the same probability. It describes the number of success in n trials.
Formula for binomial distribution
X ~ B(n, p(success))
What is P(X) for binomial
P(X) = nCr x P^X x (1 - P)^(n - x)
E(X) and V(X) for binomial
E(X) = np V(X) = np(1-p)
What is a poison distribution? Discrete or continuous?
Number of occurrences of an event in a defined time interval. Discrete distribution.
What do we need for a Poisson distribution?
N to be large
Events in each period to be independent
What is E(X) and V(X) for poisson?
E(X) = V(X) = lambda ‘mean rate of occurrence / success in each interval’
What can we use as an approx to binomial? When?
Use poisson to approx binomial
Where n >50
P small p<0.1
Notation for uniform distribution
X ~ U(a, b)
What type of distribution is the uniform distribution?
Most basic continuous distribution, all values of X have an equal probability. Symmetric.
What is the PDF for uniform distribution?
f(X) = 1 / (b - a)
E(X) & V(X) for uniform distribution
E(X) = (b + a) / 2
V(X) = (b - a)^2 / 12
How to work out probabilities for uniform distribution
Integrate 1 / b - a w.r.t. X between the interval c and d
Or P(c < X < d) = d - c / b - a
Notation for normal distribution
X ~ N (M, sigma^2)
What values can X take for normal distribution
Between +- infinity
Relation between mean, median & mode for normal distribution
Mean = median = mode
Standard normal formula
Z = X - M / sqrt sigma^2
If X1 and X2 are normally distributed, what distribution does a linear combination of them follow?
Any linear combination of two normally distributed variables is also normal.
X1 + X2 ~ N(M1 + M2, sigma1^2 + sigma2^2) covariance are zero is independent random samples.
Possible range of values of X for chi squared
Only positive X values due to squaring
E(X) & V(X) for chi squared
E(X) = V (DOF) V(X) = 2V
Relation between chi squared and normal distribution
Z^2 = N(0, 1)^2 = chi squared 1
Chi squared with DOF = 1 is the square of the standard normal distribution
If we sum up chi squared, what is the new DOF?
DOF of linear combination = sum of individual DOF
Formula for F distribution
Fm,n = (chi squared m/m) / (chi squared n/n)
Relation between F and chi squared
F = ratio of 2 independent chi squared distributions scaled by their DOFs
E(X) & V(X) for F distribution
E(X) = n/(n-2) V(X) = 2n^2 (m + n - 2) / m(n - 2)^2 (n - 4)
As n tends towards infinity (DOF of denominator), what happens to F distribution?
As n to infinity, F collapses onto a single chi squared m / m (the numerator)
Formula for t distribution
T ~ N(0, 1) / sqrt chi squared n / n
What happens to a t distribution as n tends towards infinity?
T collapses onto N(0, 1) I.e. Standard normal distribution
E(X) and V(X) for t distribution
E(X) = 0 V(X) = n / (n - 2)
Relation between t distribution and F distribution
Square of t = F distribution when M=1 (when numerator of F is a chi squared 1)
