Section 5.1 The Probability Distribution for a Discrete Random Variable January 9, 2014 1
5 Some Important Discrete
Probability Distributions
5.1 The Probability Distribution for a
Discrete Random Variable
What are some continuous random variables?
Discrete distributions
Example: (Discrete case) Roll a (not necessarily fair) six-sided die once. The
possible outcomes are the faces (i.e., dots) on the die.
⑦ ⑦
⑦
⑦
⑦
⑦
⑦ ⑦
⑦ ⑦
⑦ ⑦
⑦
⑦ ⑦
⑦ ⑦ ⑦
⑦ ⑦ ⑦
A different die might have six colors for the six sides (like a Rubik’s cube).
✷
Example: (Discrete case) Toss a (not necessarily fair) coin once.
✷
Section 5.1 The Probability Distribution for a Discrete Random Variable January 9, 2014 2
Definition: The probability distribution of a discrete random variable
X consists of the possible values of X along with their associated probabilities.
We sometimes use the terminology population distribution.
Example: Fair (six-sided) die. Let X be the numerical outcome. Determine the
probability distribution of X, graph this probability distribution, and compute the
mean of the probability distribution.
Example: Toss a fair coin 3 times. Let X = number of heads.
Let Y = number of matches among the three tosses.
Section 5.1 The Probability Distribution for a Discrete Random Variable January 9, 2014 3
✷
Terminology: The expected value of X is the mean of a random variable X.
Example: (hypothetical) Suppose an airline often overbooks flights, because past
experience shows that some passengers fail to show.
Let the random variable X be the number of passengers who cannot be boarded
because there are more passengers than seats.
x P (x)
0 0.6
1 0.2
2 0.1
3 0.1
sum 1
(a) Compute the expected value of X.
(b) Suppose each unboarded passenger costs the airline $100 in a ticket voucher.
Compute the average cost to the airline per flight in ticket vouchers.
✷
Brief review of means and standard deviations
The mean, µ, of a random variable is the average of all outcomes in the population,
and is the limiting value of X̄ as n gets large.
The standard deviation, σ, of a random variable measures the spread of all
outcomes in the population, and is the limiting value of s as n gets large.
Section 5.3 Binomial Distribution January 9, 2014 4
The variance, σ2, of a random variable also measures the spread in the population,
and is the limiting value of s2 as n gets large.
σ is more intuitive than σ2, partly because σ has the same units as the original data.
5.3 Binomial Distribution
Example: Toss an unfair coin 5 times, where π = P (heads) = 0.4.
Let X be the number of heads.
Suppose we want to determine P (X = 2).
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Factorials: m! = m(m− 1)(m− 2) · · · 1 for positive integers m.
Example: Compute 4!, 5!, 1!, and 0!.
✷
A Bernoulli trial can have two possible outcomes, success or failure.
Definition of a binomial random variable X.
1. Let n, the number of Bernoulli trials, be fixed in advance.
2. The Bernoulli trials are independent.
3. The probability of success of a Bernoulli trial is π, which is the same for all
observations.
Section 5.3 Binomial Distribution January 9, 2014 5
Let X be the number of successes. Then, X is a binomial(n, π) random variable.
P (X = x) =n!
x! (n− x)!πx (1− π)n−x, for x = 0, 1, 2, . . . , n
Example: n = 5 tosses of an unfair coin.
Assume π = P (heads) = 0.4 (OR 40% Democrats from a huge population).
Let X be the number of heads.
Determine the probability distribution of X and construct the line graph.
x P (x)
0 0.0776
1 0.2592
2 0.3456
3 0.2304
4 0.0768
5 0.01024
0 1 2 3 4 5
00.0
50.1
0.150
.20.2
50.3
0.35
X
PROB
ABILIT
Y
Determine the probability of obtaining at least one heads among the five coin tosses.
In 100 tosses of this coin, on average, how many heads do you expect?
✷
Mean and standard deviation of a binomial random
Section 5.3 Binomial Distribution January 9, 2014 6
variable
µ = nπ, σ2 = nπ(1− π), σ =√
nπ(1− π)
Example: Revisit. Let X ∼ Binomial(n = 5, π = 0.4). Compute the mean and
standard deviation of X.
✷
Example: Consider a huge population where 30% of the people are Democrats.
Let X be the number of Democrats in a sample of size 1000. Compute the mean
and standard deviation of X.
For large sample sizes (i.e., nπ ≥ 5 and n(1− π) ≥ 5), a binomial random variable and
a sample proportion are approximately normally distributed by the Central
Limit Theorem.
✷
Example: Viewing the Central Limit Theorem.
(a) Consider the graphs below for binomial random variables, using π = 0.3
and n = 1, 2, 3, 4, 5, 10, 15, 20, and 30.
Section
5.3Binom
ialDistribu
tionJanuary
9,2014
7
01
0.0 0.2 0.4 0.6
X
PROBABILITY
01
2
0.0 0.1 0.2 0.3 0.4 0.5
X
PROBABILITY
01
23
0.0 0.1 0.2 0.3 0.4
X
PROBABILITY
01
23
4
0.0 0.1 0.2 0.3 0.4
X
PROBABILITY
01
23
45
0.0 0.1 0.2 0.3X
PROBABILITY0
24
68
10
0 0.05 0.1 0.15 0.2 0.25
X
PROBABILITY
05
10
15
0 0.05 0.1 0.15 0.2
X
PROBABILITY
05
10
15
20
0 0.05 0.1 0.15
X
PROBABILITY
05
10
15
20
25
30
0.00 0.05 0.10 0.15
X
PROBABILITY
(b)Con
sider
thegrap
hsbelow
forsample
pro
portio
ns,p,
usin
gπ=
0.3
andn=
1,2,
3,4,
5,10,
15,20,
and30.
Section 5.3 Binomial Distribution January 9, 2014 8
0 1
0.00.2
0.40.6
n = 1, and Pi = 0.3
p
PROB
ABILI
TY
0 0.5 1
0.00.1
0.20.3
0.40.5
n = 2, and Pi = 0.3
p
PROB
ABILI
TY
0 0.2 0.4 0.6 0.8 1
0.00.1
0.20.3
0.4
n = 3, and Pi = 0.3
p
PROB
ABILI
TY
0 0.2 0.4 0.6 0.8 1
0.00.1
0.20.3
0.4
n = 4, and Pi = 0.3
p
PROB
ABILI
TY
0 0.2 0.4 0.6 0.8 1
0.00.1
0.20.3
n = 5, and Pi = 0.3
p
PROB
ABILI
TY0 0.2 0.4 0.6 0.8 1
00.0
50.1
0.15
0.20.2
5
n = 10, and Pi = 0.3
p
PROB
ABILI
TY
0 0.2 0.4 0.6 0.8 1
00.0
50.1
0.15
0.2
n = 15, and Pi = 0.3
p
PROB
ABILI
TY
0 0.2 0.4 0.6 0.8 1
00.0
50.1
0.15
n = 20, and Pi = 0.3
p
PROB
ABILI
TY
0 0.2 0.4 0.6 0.8 1
00.0
50.1
0.15
n = 30, and Pi = 0.3
p
PROB
ABILI
TY
✷
Example: The Democrats.
(a) Use the 95% part of the empirical rule on the binomial random variable.
00.0
050.0
10.0
150.0
20.0
25
Probability is 0.95
X
probab
ility de
nsity
functio
n
271 300 329
(b) Use the 95% part of the empirical rule on the sample proportion.
Section 5.4 Poisson Distribution January 9, 2014 9
05
1015
2025
Probability is 0.95
p
probab
ility de
nsity
functio
n0.271 0.3 0.329
✷
Read p. 194, Microsoft Excel.
5.4 Poisson Distribution
Consider the Binomial(n, π) distribution, such that n is huge, π is small, but nπ is
moderate (i.e., neither huge nor small).
Example: Radioactive decay. Consider a radioactive substance containing
3,000,000 atoms, such that decaying atoms are independent of each other, and
π = P (A particular atom decays in the next day) = 1/1, 000, 000.
Compute the mean number of atomic decays in the next day.
✷
Consider letting n → ∞ and π → 0 such that nπ → λ, a positive constant, where
X ∼ Binomial(n, π).
In this limit, X ∼ Poisson(λ).
If X ∼ Poisson(λ) for λ > 0, then
P (X = x) =1
x!λx e−λ, for x = 0, 1, 2, . . .
Section 5.4 Poisson Distribution January 9, 2014 10
Example: Revisit radioactive decay. Consider a radioactive substance containing
3,000,000 atoms, such decaying atoms are independent of each other, and
π = P (A particular atom decays in the next day) = 1/1, 000, 000.
(a) Let X1 be the number of decays in one day. Determine the probability that
at least one atom decays in the next day.
(b) Let X2 be the number of decays in two days. Determine the probability that
at least one atom decays in the next two days.
0 2 4 6 8
00.0
50.1
0.15
0.2
Poisson( λ1 = 3 )
X1
prob
abilit
y mas
s fun
ction
0 5 10 15
00.0
50.1
0.15
Poisson( λ2 = 6 )
X 2
prob
abilit
y mas
s fun
ction
✷
Remark: Typically, when a Binomial(n, π) distribution is reasonably
approximated by a Poisson(λ) distribution, n and π are difficult to determine (or
estimate), but λ can be estimated from the data. How?
The Poisson Process is explained by the following:
(a) The probability of a success (such as a radioactive decay) in the next day is
independent of its past.
(b) The mean of a Poisson process based on two days is twice as large as the
mean of the same Poisson process based on one day.
Section 5.4 Poisson Distribution January 9, 2014 11
Example: Consider the number of recombinations (breaks) in DNA (chromosome
pairs) when DNA strands are passed to offspring.
✷
Read p. 200, Microsoft Excel.
Read pp. 212–214, Appendix E5: Using Microsoft Excel for
Discrete Probability Distributions.