Transcript Document

Lesson 6 - R
Discrete Probability Distributions
Review
Objectives
• Understand when a probability experiment follows
one of the processes for discrete random variables:
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Uniform (can be continuous as well)
Binomial
Geometric
Poisson
Hypergeometric
Negative Binomial
• Compute probabilities of discrete random variables
• Find the mean and standard deviation of a discrete
random variables
Vocabulary
• None new
English Phrases
Math
Symbol
≥
At least
>
More than
<
Fewer than
≤
No more than
=
Exactly
≠
Different from
English Phrases
No less than
Greater than
Less than
At most
Equals
Greater than or equal to
Less than or equal to
Is
∑P(x) = 1
P(X)
Cumulative
probability
or cdf
P(x ≤ A)
Values of Discrete Variable, X
cdf(x > A) = 1 – P(x ≤ A)
X=A
Discrete PDF Rules
Let P(x) denote the probability that the random
variable X equals x, then
1) The sum of all probabilities of all outcomes must
equal 1
∑ P(x) = 1
2) The probability of any value x, P(x), must
between 0 and 1
0≤ P(x) ≤ 1
Mean & Variance
Mean value of a Discrete Random Variable:
Given by the formula
μx = ∑ [x ∙P(x)]
(Expected Value = μx)
where x is the value of the random variable and
P(x) is the probability of observing x
Variance and Standard Deviation of a Discrete Random Variable:
The variance of a discrete random variable is given by:
σ2x = ∑ [(x – μx)2 ∙ P(x)] = ∑[x2 ∙ P(x)] – μ2x
and standard deviation is √σ2
Discrete PDFs
Uniform: probability is equal distributed
Binomial: number of successes in n trials
Geometric: number of trials until a success or failure is observed
Hypergeometric: sampling without replacement in a small
population
Negative Binomial: number of trials necessary to observe r# of
successes (Geometric is a special case with r=1)
Poisson: number of successes in a interval of fixed length (time,
distance, or area)
Binomial Experiment Criteria
1) Experiment is performed a fixed number of times.
Each repetition is called a trial.
2) The trials are independent
3) For each trial there are two mutually exclusive
(disjoint) outcomes: success or failure
4) The probability of success is the same for each trial
of the experiment
Binomial Notation
The probability of obtaining x successes in n
independent trials of a binomial experiment, where the
probability of success is p, is given by:
P(x) = nCx px (1 – p)n-x,
x = 0, 1, 2, 3, …, n
Where n is # of independent trials of the experiment
p denotes the probability of success
1 – p is the probability of failure
x denotes the number of successes in n
independent trials of the experiment. So 0 ≤ x ≤ n
Determining probabilities using calculator: 2nd VARS:
binompdf(n,p,x) or binomcdf(n,p,x)
Binomial PDF and Law of Large Numbers
As the number of trials n in a binomial experiment
increases, the probability distribution of the random
variable X becomes bell shaped.
As a rule of thumb, if np(1-p) ≥ 10, the probability
distribution will be approximately bell shaped. (which
means the Empirical Rule can be used!!)
Geometric Criteria
An experiment is said to be a geometric experiment
provided:
1. Each repetition is called a trial.
2. The trials are independent
3. For each trial there are two mutually exclusive (disjoint)
outcomes: success or failure
4. The probability of success is the same for each trial of
the experiment
Geometric Notation
The geometric distribution addresses the number of
trials necessary before the first success. If the trials
are repeated k times until the first success, we will
have had k – 1 failures. If p is the probability for a
success and q (1 – p) the probability for a failure, the
probability for the first success to occur at the kth trial
will be (where x = k)
P(x) = p(1 – p)x-1,
x = 1, 2, 3, …
Using the calculator: 2nd VARS geometpdf(p,k)
for P(X=k)
Using the calculator: 2nd VARS geometcdf(p,k)
for P(k ≤ X)
Hyper-Geometric Criteria
An experiment is said to be a hyper-geometric experiment
provided:
1. The experiment is performed a fixed number of times.
2. Each repetition is called a trial.
3. The trials are dependent
4. For each trial there are two mutually exclusive (disjoint)
outcomes: success or failure
Note: One of the conditions of a binomial distribution was the
independence of the trials so the probability of a success is the same
for every trial. If successive trials are done without replacement and
the sample size or population is small, the probability for each
observation will vary.
Hyper-Geometric Notation
The formula for the hyper-geometric distribution is:
NpCx
N(1-p)Cn-x
P(x) = --------------------------NCn
x = 0, 1, 2, 3, …, n
Where
N is the size of the population,
p is the proportion of the population with a certain
attribute (success),
x is the number of individuals from the population
selected in the sample with the attribute and
n is the number selected to be in the sample (n-x is
the number selected who do not have the attribute)
Negative Binomial Criteria
An experiment is said to be a negative binomial
experiment provided:
1. Each repetition is called a trial.
2. For each trial there are two mutually exclusive (disjoint)
outcomes: success or failure
3. The probability of success is the same for each trial of
the experiment
4. The trials are independent
5. The trials are repeated until r successes are observed,
where r is specified in advance.
Negative Binomial Notation
When we studied the Binomial distribution, we were
only interested in the probability for a success or a
failure to happen. The negative binomial distribution
addresses the number of trials necessary before the rth
success. If the trials are repeated x times until the rth
success, we will have had x – r failures. If p is the
probability for a success and (1 – p) the probability for
a failure, the probability for the rth success to occur at
the xth trial will be
P(x) =
x-1Cr-1
pr(1 – p)x-r
x = r, r+1, r+2, …
Where r number of successes is observed in x
number of trials of a binomial experiment with
success rate of p
Poisson Criteria
Poisson distributions can be used to computer
probabilities of experiments in which the random
variable X counts the number of occurrences
(successes) of a particular event within a specified
interval (usually time or space).
An experiment is said to be a Poisson process provided:
1) Probability of two or more successes in any
sufficiently small sub-interval* is 0.
2) The probability of success is the same for any two
intervals of equal length
3) The number of successes in any interval is
independent of the number of successes in any other
interval provided the intervals are not overlapping
Poisson Notation
If X is the number of successes in an interval of fixed
length t, then probability formula for X is
(λt)x -λt
P(x) = --------- e
x = 0, 1, 2, 3, …
x!
where λ (the Greek letter lamda) represents the average
number of occurrences of the event in some interval of
length 1 and e = 2.71828.... (Euler's constant)
Using the calculator: Poissonpdf(μ,x) & Poissoncdf(μ,x)
Discrete Probability Functions Summary
Name
Binomial
PDF
P(x) = nCxpx(1-p)n-x
x=
x = 0,1,2,…
μ
σ
np
√np(1-p)
1/p
√1-p
Example: Coin flips (x = number of heads in n=10 trials)
Geometric
P(x) = p(1 – p)x-1
x = 1,2,…
Example: number of trials necessary before the first success
How many flips before the first tail is observed?
NpCx N(1-p)Cn-x
HyperP(x) = --------------------x = 0,1,2,…
np
geometric
NCn
√ (np(1-p)(N-n)/N-1)
Example: Small number sampling without replacement
What is P(x=6) women will be selected from population of 20 men & 20 women (N=40) in
a n=10 random sampling?
Negative
Binomial
P(x) = x-1Cr-1 pr(1 – p)x-r
x = r,r+1,…
r/p
√ (r(1-p)/p2)
Example: r number of successes in x trials (of a binomial experiment)
What is the P(x=5) flips are necessary to get 3 heads?
(λt)x
P(x) = ------- e-λt
Poisson
x = 0,1,2,…
λt
x!
Example: Events occurring over time or space
What is the P(x<15) people per hour going through McD’s drive through?
√λt
Summary and Homework
• Summary
– Probability rules remain the same
– Each of the six Discrete PDFs have criteria
that are similar, but slightly different
– Calculator has pdf and cdf functions for 3
of the 6 Discrete PDFs
• Homework
– pg 352 - 355: 1, 4, 5, 9, 13, 19