Transcript Notes – Chapter 17 Binomial & Geometric Distributions

Notes – Chapter 17
Binomial & Geometric
Distributions
Binomial Distributions
Conditions for a Binomial Distribution:
• 1) Only two categories
(i.e. success or failure)
• 2) Observations are independent*
• 3) Probability of success is the same for
each observation.
• 4) Fixed number of n observations
Binomial Distributions
*we can use the binomial distributions in the
statistical setting of selecting an SRS
when the population is much larger than
the sample without worrying about
replacement
Binomial Distributions
If the conditions for a binomial distribution
are met, then a binomial distribution can be
built. The binomial variable, X, is numeric
and is a count of the number of successes
for the given event.
Binomial Distributions
• Binomial notation is
B(n, p)
for any binomial distribution,
where n is the number of trials
and p is the probability
Binomial Distributions
• IF we are interested in a specific value of
X: P(X = ?)
PDF– Given a discrete random variable X,
the probability distribution function
assigns a probability to each value of X.
Binomial Distributions
Calculator use for binomial:
P(x = k) is calculated binompdf(n, p, k)
So…
In a binomial distribution B(20, .4),
P(x = 5) is calculated binompdf(20, .4, 5)
Binomial & Geometric
Distributions
• If we are interested in a range of values for X:
P(X ≤ ?)
CDF – Given a random variable X, the cumulative
distribution function of X calculates the sum of
the probabilities for 0, 1, 2, … up to the value of
X. That is, it calculates the probability of
obtaining at most X successes in n trials.
• (See calculator use on page 393 for more info)
Binomial Distributions
• P(x ≤ k) is calculated binomcdf(n, p, k)
So…
In a binomial distribution B(20, .4),
P(x ≤ 5) is calculated binomcdf(20, .4, 5)
Binomial & Geometric
Distributions
• P(x < k), P(x > k) and P(x ≥ k) can not be
calculated directly from the calculator.
You must rephrase the inequality in terms of
one of the previous two statements and
sometimes use the complement rule.
Binomial Distributions
• Binomial Coefficient
• The number of ways I can arranging k
successes among n observations when
order matters is given by the binomial
coefficient
• for k = 0, 1, 2, …, n
Binomial Distributions
• Binomial Probability
• If X has a binomial distribution B(n, p)
then…
n k
nk
P( x  k )    p (1  p)
k
Binomial Distributions
• Mean and Standard Deviation of a
Binomial Distribution
 = np
 = np(1 – p)
Geometric Distributions
• The Geometric Distribution
Used when the goal is to obtain the first
success. A random variable X can be
defined that counts the number of trials
needed to obtain that first success.
Geometric Distributions
• Conditions for a Geometric Distribution
1) Only two categories
(i.e. success or failure)
2) Observations are independent*
3) Probability of success is the same for
each observation.
4) The variable of interest is the number
of trials required to obtain the first success.
Geometric Distributions
*we can use the geometric distributions in
the statistical setting of selecting an SRS
when the population is much larger than
the sample without worrying about
replacement
Geometric Distributions
• Rules for calculating Geometric
Probabilities:
P(X = n) = (1 – p)n-1p
P(X > n) = (1 – p)n
Geometric Distributions
• The mean of a Geometric Random
Variable
 = 1/p
Geometric Distributions
• The calculator has a geometpdf and
geometcdf that functions like their binomial
friends. If you want to use them you may
but you still have to show your work for
credit on any test (including the AP).
Binomial & Geometric
Distributions
Showing your work….
1) Always check your conditions before using
either formula. Be specific & use context.
2) Always show the numbers you used for
Geometric distributions. You do not need to write
the Geometric formulas.
Binomial & Geometric
Distributions
Showing your work….
3) Substitute into the binomial formulas
for P(x = k) only not for <, > etc...
4) Expand and/or rewrite the probability
statements for any binomials inequalities.
Do not substitute into the binomial
formula.