Statistical Sciences 2244A/B Lecture Notes - Lecture 5: Binomial Distribution, Random Variable, Simple Random Sample
22 May 2018
School
Department
Professor
Stats 2244
Random Variables
Discrete RVs, Binomial Distributions
Random Variables (RV)
- Most of the things in this course assumes we are dealing with a random variable
- A variable which has a numerical outcome of a random phenomenon (ie procedure)
- Example: let X be the RV equal to the # of student in an SRS of 4 who have a job. Assume having
a job is equally likely as not having a job
o The outcomes are the # of students who have a job – which ranges bw 0-4
o The random phenomenon here is the SRS that defines the individuals and the outcomes
are the # of those students who have a job
o Probabilities for the outcomes:
▪ If we have 4 students and none of them have a part time job, itll be NNNN
▪ But if we have 1 of the 4 students with a part time job, then there are 4 simple
events that define 1 out of 4 students having a part time job
o We could use the fact that there are 16 possible combinations,
▪ meaning that the probability of get 2 student with a part time job is 6/16
▪ probability of having no students with a job and all students with a job is equally
likely → 1/16
o does this example meet the criteria of a binomial distribution?
▪ N is fixed and outcomes being only classified into 2 categories (success and
failure) is met
▪ SRS could be with replacement and it might also not be with replacement
▪ When you see SRS, assume its WITH replacement, unless told otherwise
▪ This means, probability being constant is not an issue bc you take an individual
out, see if they have a job or not then put them back into the population
▪ SRS should clean up the concerns associated with the independence of trials (bc
we are just selecting ppl randomly) and the probability being constant issue
▪ So YES it is a binomial distribution
Binomial Probability Model
- Binomial distribution is a type of discrete random variable
If a random variable (X) meets the following conditions, then the probability of a particular outcome, x,
is:
- This probability model for binomial distributions defines how likely it is a for an outcome to
happen
- Tells us that X is = to one of its possible outcomes (x)
- The (n!)/((n-x)!x! tells us the number of possible combinations
o This can also be written as nCx or nCr or but x instead of k
o Counts the number of possible combinations in x successes out of n trials
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Most of the things in this course assumes we are dealing with a random variable. A variable which has a numerical outcome of a random phenomenon (ie procedure) Example: let x be the rv equal to the # of student in an srs of 4 who have a job. Binomial distribution is a type of discrete random variable. If a random variable (x) meets the following conditions, then the probability of a particular outcome, x, is: This probability model for binomial distributions defines how likely it is a for an outcome to happen. Tells us that x is = to one of its possible outcomes (x) The (n!)/((n-x)!x! tells us the number of possible combinations: this can also be written as ncx or ncr or, counts the number of possible combinations in x successes out of n trials but x instead of k.
