ECON1203 Lecture Notes - Lecture 5: Bernoulli Trial, Binomial Distribution, Random Variable
4 – Probability cont.
Binomial distribution
• Based o otio of a ioial eperiet osistig of a sequence of trials
• Requirements/assumptions to qualify as a binomial experiment:
Sequence of fixed number of n trials
Each trial has two outcomes, aritraril deoted suess ad failure
Fixed probability of success, p, over all trials
Trials are independent
Under these assumptions, this is a sequence of Bernoulli trials
The outcome of each trial is recorded in a random variable
Binomial random variables
• X1, X2, …, Xn where Xi = is alled success’ and Xi = is alled failure’
• Under the assumptions made, this is a sequence of independent and identically
distributed (iid) random variables
• We can form another random variable from these Bernoulli random variables
Consider the random variable formed by summing these n Bernoulli random
variables: X = X1 + X2 + … + Xn
X represents the number of successes in n trials
X is called binomial random variable
(a) Characterised by 2 parameters: n and p
(b) Once known, parameters tell everything about random variable and its
probability distribution
Binomial distribution
• Combinatorial formula: how many distinct combinations exist when we choose x
objects from n objects, without regard to order?
• The probability of observing x successes in a binomial experiment with n trials and
probability of success p, is:
Members of this family of distributions are characterised by n and p
Different binomial random variables are generated with different n and p
combinations
Binomial tables
• Tables provide entire distribution
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Binomial distribution: based o(cid:374) (cid:374)otio(cid:374) of a (cid:858)(cid:271)i(cid:374)o(cid:373)ial e(cid:454)peri(cid:373)e(cid:374)t(cid:859) (cid:272)o(cid:374)sisti(cid:374)g of a sequence of trials, requirements/assumptions to qualify as a binomial experiment: Sequence of fixed number of n trials. Each trial has two outcomes, ar(cid:271)itraril(cid:455) de(cid:374)oted (cid:858)su(cid:272)(cid:272)ess(cid:859) a(cid:374)d (cid:858)failure(cid:859) Fixed probability of success, p, over all trials. Under these assumptions, this is a sequence of bernoulli trials. The outcome of each trial is recorded in a random variable. Consider the random variable formed by summing these n bernoulli random variables: x = x1 + x2 + + xn. X represents the number of successes in n trials. X is called binomial random variable (a) characterised by 2 parameters: n and p (b) once known, parameters tell everything about random variable and its probability distribution. Members of this family of distributions are characterised by n and p. Different binomial random variables are generated with different n and p combinations. Mean and variance of a binomial random variable.
