Question

The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.2% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same-sex couples should have the right to legal marital status. Construct the probability distribution function (PDF).

Answer 1

N = total number of students =203967

n = number of students randomly selected.= 8

p = probability of success

= P ( Students believes that same-sex couples should have the right to legal marital status)

From the information

p = 71.2% = 0.712

Let X denotes number of students believes that same-sex couples should have the right to legal marital status.

X takes value 0,1,2,.........8 and probability of success (0.712) is constant for each students.

Hence the random variable X follows binomial distribution with parameter n =8 and p = 0.712.

i.e. X ~ Bin(n=8,p=0.712)

The probability distribution function of random variable X is

and Cumulative density function is

By using R

> p=0.712
> n=8
> x=0:n
> prob=dbinom(x,n,p)
> p1=round(prob,5)
> cumprob=pbinom(x,n,p)
> p2=round(cumprob,5)
> d=data.frame("X"=x,"Probability" = p1, "Cumulative probability"=p2)
> d
X Probability Cumulative.probability
1 0 0.00005 0.00005
2 1 0.00094 0.00098
3 2 0.00810 0.00908
4 3 0.04005 0.04913
5 4 0.12376 0.17289
6 5 0.24477 0.41767
7 6 0.30257 0.72024
8 7 0.21372 0.93395
9 8 0.06605 1.00000

from the output

The probability distribution of random variable X is

x	0	1	2	3	4	5	6	7	8
P(X=x)	0.00005	0.00094	0.00810	0.04005	0.12376	0.24477	0.30257	0.21372	0.06605