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Overview
You should be able to:
Identify a random variable as a binomial or geometric
Use the formula to determine binomial or geometric probabilities
Calculate cumulative distribution functions for binomial random variables and geometric random variables and construct cumulative distribution tables and histograms
Calculate means, expected values, and standard deviations of binomial and geometric variables
Use Normal approximation to the binomial distribution to compute probabilities
Binomial vs. Geometric Variables
Counts the number of successes X in a given number of trials n
Counts the number of trials X needed to get the first success
Binomial Geometric
Both are discrete random variables.
Binomial Setting
Each observation falls into one of two categories: success or failure
Observations are all independent: knowing the result of one observation tells you nothing about the others
Probability of success (p) is the same for each observation
Fixed number (n) of observations
Variables and Parameters
X= binomial random variable that is the number of successes
Probability distribution of X is a binomial distribution
Parameters:
n= number of observations
p= probability of success of any one observation
0<X<n, whole numbers only
Sampling Distribution
choose an SRS of size n from the population with proportion p of successes
when population is at least 10 times larger than the sample (N>10n), X number of successes in sample has approximate binomial distribution with parameters n and p
X = B(n,p)
Binomial Coefficient
Number of ways of arranging k successes among n observations
Note: not n divided by k, but n choose k
nkknk
n
k
n0,
)!(!
!
Binomial Probability
P(X=k)
Binomial coefficient X probability of any arrangement of k success
X=# of successes
n=# of observations
p= probability of success each time
k= a value of x
)1()( ppknk
k
nkXP
pdf vs. cdf
Probability distribution function (pdf)
assigns a probability to each value of X
Cumulative distribution function (cdf)
calculates the sum of the probabilities for 0,1,2…. up to the value of X, including the value of X
calculates the probability of obtaining at most X successes in n trials
Geometric Setting
Each observation falls into one of two categories: success or failure
The observations are independent
The probability of success (p) is the same for each observation
The variable of interest (X) is the number of trials required to obtain the first success
Calculator Keystrokes
To… Do This…
Calculate binomial probability of X 2nd→vars→A: binompdf(
P(X=k)=binompdf(n,p,X)
Calculate binomial probability up to or
equal to X
2nd→vars→A: binomcdf(
P(X=k)=binomcdf(n,p,X)
Simulate many trials in a binomial setting Math→PRB→7: randBin(
randBin(1,p,n)
1=success; 0=failure
Count the number of successes in n trials Simulate many trials
STO→L1: sum (L1)
Execute Math→PRB→3: nCr
n nCr k
Calculate geometric probability 2nd→VARS→E:geometpdf(
geometpdf(p of failing, n)
k
n