STAT1008 Lecture Notes - Lecture 21: Binomial Coefficient, Binomial Distribution, Random Variable
STAT1008 Week 7 Lecture C
● Binomial Coefficient:
○ The number of arrangements of k successes among n trials can be computed
with a binomial coefficient
■ Literally just 3u Maths holy cow
■ n!/k!(n-k)! = nCk
● Binomial Probability Function:
○ For a binomial random variable with n trials and probability of success p on each
trial, the probability of exactly k successes in the n trials is:
■ P (X=k) = (n k)pk(1-p)n-k
● Using R Studios:
○ Function to use pbinom and qbinom
○ pbinom (x, size, prob, lower.tail=TRUE)
○ qbinom (x, size, prob, lower.tail=TRUE)
○ P (X>7) (At least 8 free throws out of 10) thus pbinom (7,10,0.9,
lower.tail=FALSE) = [1].0.9298
● Mean for Binomial:
○ For a binomial random variable X with n trials and probability of success p on
each trial, the mean is
■ E(X) = mu = n.p
■ Stn Dev = sqrt(np(1-p))
● Some bootstrap and randomisation distributions
○ All bell-shaped distributions
● Density curve:
○ A density curve is a theoretical model to describe a variable’s distribution
○ Think of a density curve as an idealised histogram, where:
■ The total area under the curve is one
■ The proportion of the population in any interval is the area over that
interval
● Normal Distribution:
○ A normal distribution has a symmetric bell shaped density curve
○ Two features distinguish one normal density from another:
■ The mean is its center of symmetry (mu)
■ The standard deviation controls its spread
● Calculate normal distribution things:
○ STATKEY:
■ Go to statkey and put the values for mean and standard deviation and
can customise the cutoff for the left tail and right tails
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
The number of arrangements of k successes among n trials can be computed with a binomial coefficient. For a binomial random variable with n trials and probability of success p on each trial, the probability of exactly k successes in the n trials is: P (x>7) (at least 8 free throws out of 10) thus pbinom (7,10,0. 9, lower. tail=false) = [1]. 0. 9298. For a binomial random variable x with n trials and probability of success p on each trial, the mean is. A density curve is a theoretical model to describe a variable"s distribution. Think of a density curve as an idealised histogram, where: The total area under the curve is one. The proportion of the population in any interval is the area over that interval. A normal distribution has a symmetric bell shaped density curve. Two features distinguish one normal density from another: The mean is its center of symmetry (mu)
