Degenerate ,Bernoulli , Binomial Distribution
Degenerate distribution (one point distribution):
P(X = c) = 1, mean and variance.
Uniform discrete distribution on integers 1 to n: p.m.f., c.d.f.,
mean, variance, real life situations, comments on mode and median.
Bernoulli Distribution: p.m.f., mean, variance.
Binomial Distribution: p.m.f.
=Qp*q"
P(x) =
= 0
p* q, x = 0, 1, 2,, ...; 0 < p < 1, q = 1-p
Notation: X B(n, p).
; otherwise
Recurrence relation for successive probabilities, computation of probabilities of different events, mode of the distribution, mean, variance, m.g.f. and c.g.f. moments, skewness (comments when p = 0.5, p > 0.5, p < 0.5). Situations where this distribution is applicable.
Additive property for binomial distribution.
Conditional distribution of X given (X + Y) for binomial distribution.
Necessity and importance of Hypergeometric distribution, capture-recapture method.
p.m.f. of the distribution,
MM-M)
Μ
P(x) =
, x = 0, 1, ..., min (M, n)
= 0,
Notation X ~ H(N, M, n).
otherwise
Computation of probability, situations where this distribution is applicable, binomial approximation to hypergeometric probabilities, statement of mean and variance of the distribution (Derivation is not expected).
p.m.f. of the distribution
emm* p(x) = x!
x = 0, 1, 2, ..., m > 0
= 0
;
otherwise
Notation: X~ P(m).
m.g.f. and c.f.g.f moments, mean, variance, skewness and kurtosis.
Situations where this distribution is applicable.
Additive property for Poisson distribution.
Conditional distribution of X given (X + Y) for Poisson distribution.
Notation: X~ G(p),
Geometric distribution on support (0, 1, 2, ...) with p.m.f. p(x) = pq. Geometric distribution on support (1, 2, ...) with p.m.f. p(x) = pq*-1, 0 < p < 1, q1 p.
Mean, variance, m.g.f. and c.g.f.
Situations where this distribution is applicable.
Lack of memory property.
Negative Binomial Distribution:
Probability mass function (p.m.f.)
P(X = x) =
+k - 1 X
-1)p* q Jp
; x = 0, 1, 2, ...
0
Notation: N~ NB(k, p).
0<p<1;q=1- p;k > 0
; otherwise
Graphical nature of p.m.f., negative binomial distribution as a waiting time distribution, moment generating function (M.G.F.), cumulant generating function (CGF), mean, variance, skewness, kurtosis (recurrence relation between moments is not expected), additive property of NB(k, p). Relation between geometric distribution and negative binomial distribution. Poisson approximation to negative binomial distribution. Real life situations.
Concept of truncated distribution, truncation to the right, left and on both sides. Binomial distribution left truncated at X = 0 (value zero is discarded), its p.m.f., mean and variance. Poisson distribution left truncated at X = 0 (value zero is discarded), its p.m.f., mean and variance. Real life situations and applications.