HCMC University of Technology
Probability and Statistics
Dung Nguyen
Some Special Distributions
Outline I
1
Discrete Distributions
2
Continuous Distributions
Dung Nguyen
Probability and Statistics 2/44
Discrete distributions
Binomial distributions B(n, p)
Poisson distributions Poisson(λ)
Continuous distributions
Continuous uniform distributions U(a, b)
Exponential distributions Exp(λ)
Normal distributions N(m, σ 2 )
The central limit theorem (CLT)
Dung Nguyen
Probability and Statistics 3/44
Discrete Distributions
1
Discrete Distributions
The Binomial Distributions B(n, p)
The Poisson Distributions Poisson(λ)
Dung Nguyen
Probability and Statistics 4/44
The Binomial Distributions B(n, p)
Discrete Distributions
The Binomial Distributions B(n, p)
Definition
Bernoulli trial B(p):
u
1
0
P(Z = u) p q = 1 − p
E(Z) = p and V(Z) = pq.
Binomial random variable = the # of successes in n Bernoulli
trials (the probability of success in each trial is 0 ≤ p ≤ 1).
Examples:
The # of defective items among 20 independent items with the
defective rate 5%.
The # of winning tickets among 11 independent lottery tickets
with the winning rate 1%.
The # of patients reporting symptomatic relief with a specific
medication with the effective rate 80%.
Dung Nguyen
Probability and Statistics 5/44
The Binomial Distributions B(n, p)
Discrete Distributions
Example 1 - Is X a binomial random variable?
a
A coin is weighted in such a way so that there is a 70% chance
of getting a head on any particular toss. Toss the coin, in
exactly the same way, 100 times. Let X equal the number of
heads tossed.
b
A college administrator randomly samples students until he
finds four that have volunteered to work for a local
organization. Let X equal the number of students sampled.
c
A Quality Control Inspector (QCI) investigates a lot containing
15 skeins of yarn. The QCI randomly samples (without
replacement) 5 skeins of yarn from the lot. Let X equal the
number of skeins with acceptable color.
d
A Gallup Poll of n = 1000 random adult Americans is conducted.
Let X equal the number in the sample who own a sport utility
vehicle (SUV).
Dung Nguyen
Probability and Statistics 6/44
The Binomial Distributions B(n, p)
Discrete Distributions
n = 30, p = 0.5
n = 30, p = 0.2
n = 8, p = 0.5
n = 8, p = 0.2
0.3
0.15
0.2
0.1
0.1
0.05
0
0
0
2
4
6
8
0
5
10
Figure: Pmf of B(n, p)
Dung Nguyen
Probability and Statistics 7/44
15
20
25
30
The Binomial Distributions B(n, p)
Discrete Distributions
Proposition
Let X ∼ B(n, p).
1
Then
X takes values in Ω = {0, 1, . . . , n} such that
f(k) = P(X = k) = Ckn pk q n−k .
k = 0:
k = n:
k = 1:
k = 2:
2
q
p
p
p
q
p
q
p
q
p
q
q
···
···
···
···
q
p
q
q
=⇒
=⇒
=⇒
=⇒
P(X = 0) = q n .
P(X = n) = pn .
P(X = 1) = C1n pq n−1 .
P(X = 2) = C2n p2 q n−2 .
X is a sum of n independent Bernoulli random variables.
X = Z1 + Z2 + · · · + Zn ,
3
(
1,
where Zi =
0,
if the i-th trial is successful
.
otherwise
E(X) = np
V(X) = npq.
and
Dung Nguyen
Probability and Statistics 8/44
The Binomial Distributions B(n, p)
Discrete Distributions
Example
2
Each sample of water has a 10% chance of containing a
particular organic pollutant. Assume that the samples are
independent with regard to the presence of the pollutant.
Dung Nguyen
Probability and Statistics 9/44
The Binomial Distributions B(n, p)
Discrete Distributions
Example
2
Each sample of water has a 10% chance of containing a
particular organic pollutant. Assume that the samples are
independent with regard to the presence of the pollutant.
a
Find the probability that, in the next 18 samples, exactly 2
contain the pollutant.
Dung Nguyen
Probability and Statistics 9/44
The Binomial Distributions B(n, p)
Discrete Distributions
Example
2
Each sample of water has a 10% chance of containing a
particular organic pollutant. Assume that the samples are
independent with regard to the presence of the pollutant.
a
b
Find the probability that, in the next 18 samples, exactly 2
contain the pollutant.
Determine the probability that at least 4 samples contain the
pollutant.
Dung Nguyen
Probability and Statistics 9/44
The Binomial Distributions B(n, p)
Discrete Distributions
Example
2
Each sample of water has a 10% chance of containing a
particular organic pollutant. Assume that the samples are
independent with regard to the presence of the pollutant.
a
b
c
Find the probability that, in the next 18 samples, exactly 2
contain the pollutant.
Determine the probability that at least 4 samples contain the
pollutant.
Now determine the probability that 3 ≤ X < 7.
Dung Nguyen
Probability and Statistics 9/44
The Binomial Distributions B(n, p)
Discrete Distributions
Example
2
Each sample of water has a 10% chance of containing a
particular organic pollutant. Assume that the samples are
independent with regard to the presence of the pollutant.
a
b
c
3
Find the probability that, in the next 18 samples, exactly 2
contain the pollutant.
Determine the probability that at least 4 samples contain the
pollutant.
Now determine the probability that 3 ≤ X < 7.
A certain electronic system contains 10 components. Suppose
that the probability that each individual component will fail
is 0.2 and that the components fail independently of each
other. Given that at least one of the components has failed,
what is the probability that at least two of the components
have failed?
Dung Nguyen
Probability and Statistics 9/44
The Binomial Distributions B(n, p)
Discrete Distributions
Example
4
A certain binary communication system has a bit-error rate of
0.1; i.e., in transmitting a single bit, the probability of
receiving the bit in error is 0.1. If 6 bits are transmitted,
then how many bits, on average, will be received in error?
Determine the corresponding variance.
Dung Nguyen
Probability and Statistics 10/44
The Binomial Distributions B(n, p)
Discrete Distributions
Example
4
A certain binary communication system has a bit-error rate of
0.1; i.e., in transmitting a single bit, the probability of
receiving the bit in error is 0.1. If 6 bits are transmitted,
then how many bits, on average, will be received in error?
Determine the corresponding variance.
5
Three men A, B, and C shoot at a target. Suppose that A shoots
three times and the probability that he will hit the target on
any given shot is 1/8, B shoots five times and the probability
that he will hit the target on any given shot is 1/4, and C
shoots twice and the probability that he will hit the target on
any given shot is 1/3. What is the expected number of times
that the target will be hit?
Dung Nguyen
Probability and Statistics 10/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Applications of Poisson Distributions
Electrical system example: the number of telephone calls
arriving in a system in 1 second, the number of wrong
connections to your phone number per day.
Astronomy example: the number of photons arriving at a
telescope in 1 microsecond.
Biology example: the number of mutations on a strand of DNA
per unit length, the number of bacteria on some surface or weed
in the field,
Management example: the number of customers arriving at a
counter or call centre in 10 minutes.
Civil engineering example: the number of cars arriving at a
traffic light in 5 minutes.
Finance and insurance example: the number of Losses/Claims
occurring in a given period of time.
Dung Nguyen
Probability and Statistics 11/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
The Poisson Distributions
Poisson r.v.
Unknown:
Known:
= the count of events that occur within an interval.
the # of trials n or the probability of success p
the average # of successes per time period λ = np.
k n!
λ
λ n−k
λk
lim P(Xn = k) = lim
1−
= e−λ · .
n→∞
n→∞ k!(n − k)!
n
n
k!
Dung Nguyen
Probability and Statistics 12/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
The Poisson Distributions
Poisson r.v.
Unknown:
Known:
= the count of events that occur within an interval.
the # of trials n or the probability of success p
the average # of successes per time period λ = np.
k n!
λ
λ n−k
λk
lim P(Xn = k) = lim
1−
= e−λ · .
n→∞
n→∞ k!(n − k)!
n
n
k!
Poisson distribution:
Ω = {0, 1, 2, . . .}
and
f(k) = P(X = k) = e−λ ·
Dung Nguyen
λk
,
k!
Probability and Statistics 12/44
x ∈ Ω.
The Poisson Distributions Poisson(λ)
Discrete Distributions
The mean and the variance of a Poisson distribution are the same.
E(X) = λ
V(X) = λ.
and
Random sample (B(5, 0.7)):
1 5 4 4 5 3 4 4 2 5
Then x = 3.7 and s2 = 1.7889.
Random sample (Poisson(3.5)):
2 7 3 0 6 5 2 1 2 3
Then x = 3.1 and s2 = 4.9889.
If Xi ∼ Poisson(λi ) and are independent then
n
X
Xi ∼ Poisson
i=1
n
X
!
λi
.
i=1
Dung Nguyen
Probability and Statistics 13/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
·10−2
λ = 0.5
0.6
λ = 20
λ=5
8
0.15
6
0.4
0.1
4
0.2
0.05
0
2
0
0
0
2
4
6
8
0
5
10
15
0
Figure: Pmf of Poisson(λ)
Dung Nguyen
Probability and Statistics 14/44
10
20
30
40
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
6
Consider an experiment that consists of counting the number of
α particles given off in a 1-second interval by 1 gram of
radioactive material. If we know from past experience that, on
average, 3.2 such α particles are given off, what is a good
approximation to the probability that no more than 2 α
particles will appear?
Dung Nguyen
Probability and Statistics 15/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
6
Consider an experiment that consists of counting the number of
α particles given off in a 1-second interval by 1 gram of
radioactive material. If we know from past experience that, on
average, 3.2 such α particles are given off, what is a good
approximation to the probability that no more than 2 α
particles will appear?
7
Flaws occur at random along the length of a thin copper wire.
Suppose that the number of flaws follows a Poisson distribution
with a mean of 2.3 flaws per mm. Find the probability of
Dung Nguyen
Probability and Statistics 15/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
6
Consider an experiment that consists of counting the number of
α particles given off in a 1-second interval by 1 gram of
radioactive material. If we know from past experience that, on
average, 3.2 such α particles are given off, what is a good
approximation to the probability that no more than 2 α
particles will appear?
7
Flaws occur at random along the length of a thin copper wire.
Suppose that the number of flaws follows a Poisson distribution
with a mean of 2.3 flaws per mm. Find the probability of
a
exactly 2 flaws in 1 mm of wire.
Dung Nguyen
Probability and Statistics 15/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
6
Consider an experiment that consists of counting the number of
α particles given off in a 1-second interval by 1 gram of
radioactive material. If we know from past experience that, on
average, 3.2 such α particles are given off, what is a good
approximation to the probability that no more than 2 α
particles will appear?
7
Flaws occur at random along the length of a thin copper wire.
Suppose that the number of flaws follows a Poisson distribution
with a mean of 2.3 flaws per mm. Find the probability of
a
b
exactly 2 flaws in 1 mm of wire.
exactly 10 flaws in 5 mm of wire.
Dung Nguyen
Probability and Statistics 15/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
6
Consider an experiment that consists of counting the number of
α particles given off in a 1-second interval by 1 gram of
radioactive material. If we know from past experience that, on
average, 3.2 such α particles are given off, what is a good
approximation to the probability that no more than 2 α
particles will appear?
7
Flaws occur at random along the length of a thin copper wire.
Suppose that the number of flaws follows a Poisson distribution
with a mean of 2.3 flaws per mm. Find the probability of
a
b
c
exactly 2 flaws in 1 mm of wire.
exactly 10 flaws in 5 mm of wire.
at least 1 flaw in 2 mm of wire.
Dung Nguyen
Probability and Statistics 15/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Approximation property
Let Y ∼ Poisson(λ) and Xn ∼ B(n, pn ) with pn = λ/n. Then
P(Xn = k)
lim
= 1, ∀k = 0, 1, . . .
n→∞ P(Y = k)
0.2
B(16, 0.25)
Poisson(4)
B(100, 0.04)
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0.05
0
0
0
0.15
0.1
0
5
10
15
0
5
Figure: Poisson(λ) vs.
Dung Nguyen
10
15
0
B(n, p) with np = λ
Probability and Statistics 16/44
5
10
15
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
8
Suppose that 1 in 5000 light bulbs are defective. What is the
probability that there are at least 3 defective light bulbs in
a group of size 10000?
Dung Nguyen
Probability and Statistics 17/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
8
Suppose that 1 in 5000 light bulbs are defective. What is the
probability that there are at least 3 defective light bulbs in
a group of size 10000?
9
Suppose that the number of drivers who travel between a
particular origin and destination during a designated time
period has a Poisson distribution with parameter λ = 20. What
is the probability that the number of drivers will
Dung Nguyen
Probability and Statistics 17/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
8
Suppose that 1 in 5000 light bulbs are defective. What is the
probability that there are at least 3 defective light bulbs in
a group of size 10000?
9
Suppose that the number of drivers who travel between a
particular origin and destination during a designated time
period has a Poisson distribution with parameter λ = 20. What
is the probability that the number of drivers will
a
Be at most 10?
Dung Nguyen
Probability and Statistics 17/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
8
Suppose that 1 in 5000 light bulbs are defective. What is the
probability that there are at least 3 defective light bulbs in
a group of size 10000?
9
Suppose that the number of drivers who travel between a
particular origin and destination during a designated time
period has a Poisson distribution with parameter λ = 20. What
is the probability that the number of drivers will
a
b
Be at most 10?
Be within 2 standard deviations of the mean value?
Dung Nguyen
Probability and Statistics 17/44
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
8
Suppose that 1 in 5000 light bulbs are defective. What is the
probability that there are at least 3 defective light bulbs in
a group of size 10000?
9
Suppose that the number of drivers who travel between a
particular origin and destination during a designated time
period has a Poisson distribution with parameter λ = 20. What
is the probability that the number of drivers will
a
b
10
Be at most 10?
Be within 2 standard deviations of the mean value?
Observations (following Poisson(λ)):
3, 3, 3, 3, 1, 4, 6, 1, 2, 0
Estimate λ and P(X ≤ 2).
Dung Nguyen
Probability and Statistics 17/44
Continuous Distributions
2
Continuous Distributions
The Continuous Uniform Distributions U(a, b)
The Exponential Distribution Exp(λ)
The Normal Distributions N(m, σ 2 )
Central Limit Theorem
Dung Nguyen
Probability and Statistics 18/44
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
The Continuous Uniform Distributions U(a, b)
This distribution has pdf and cdf

 1 , x ∈ [a, b]
f(x) = b − a
and
0,
otherwise
Dung Nguyen


0,

x − a
,
F(x) =

b−a


1,
Probability and Statistics 19/44
x<a
x ∈ [a, b) .
x≥b
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
0.4
U(0, 2.5)
U(0, 5)
U(0, 10)
U(−2, 2)
0.3
0.2
0.1
0
−2
0
2
4
6
8
10
Figure: Pdf of U(a, b)
Proposition (Properties)
a+b
1 E(X) =
2
2
Dung Nguyen
V(X) =
(b − a)2
12
Probability and Statistics 20/44
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
Example
11
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
Dung Nguyen
Probability and Statistics 21/44
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
Example
11
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
a
What is the probability that the current is between 4.95 mA and
5.0 mA?
Dung Nguyen
Probability and Statistics 21/44
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
Example
11
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
a
b
What is the probability that the current is between 4.95 mA and
5.0 mA?
Calculate its mean and variance.
Dung Nguyen
Probability and Statistics 21/44
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
Example
11
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
a
b
12
What is the probability that the current is between 4.95 mA and
5.0 mA?
Calculate its mean and variance.
Buses arrive at a specified stop at 15-minute intervals
starting at 7 a.m. That is, they arrive at 7, 7:15, 7:30,
7:45, and so on. If a passenger arrives at the stop at a time
that is uniformly distributed between 7 and 7:30, find the
probability that he waits more than 10 minutes for a bus.
Dung Nguyen
Probability and Statistics 21/44
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
Example
11
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
a
b
What is the probability that the current is between 4.95 mA and
5.0 mA?
Calculate its mean and variance.
12
Buses arrive at a specified stop at 15-minute intervals
starting at 7 a.m. That is, they arrive at 7, 7:15, 7:30,
7:45, and so on. If a passenger arrives at the stop at a time
that is uniformly distributed between 7 and 7:30, find the
probability that he waits more than 10 minutes for a bus.
13
Observations (following a uniform distribution):
3.6, 3.6, 3.3, 3.9, 1.6, 4.5, 4.9, 1.4, 2.7, 1.2.
Estimate P(X ≤ 2).
Dung Nguyen
Probability and Statistics 21/44
The Exponential Distribution Exp(λ)
Continuous Distributions
The Exponential Distribution Exp(λ)
Poisson process is a process in which events occur continuously and
independently at a constant average rate λ.
The exponential distribution describes the distance/time between
events in a Poisson process.
Some applications:
(by seismologists and earth scientists) To predict the
approximate time when an earthquake is likely to occur in a
particular location.
To calculate the reliability of electronic gadgets such as a
laptop, battery, processor, mobile phone. To know an
approximate time after which the product will get ruptured.
To calculate the time duration between the passing of two
consecutive cars, thereby helping the traffic in charge to
reduce the traffic problem and to avoid collisions.
Dung Nguyen
Probability and Statistics 22/44
The Exponential Distribution Exp(λ)
Continuous Distributions
N:
the number of flaws in u millimeters of wire.
X: the length from any starting point on the wire until a flaw
is detected
If the mean number of flaws is λ per millimeter then
(λu)0
= e−λu .
P(X > u) = P(N = 0) = e−λu
0!
Therefore, P(X ≤ u) = 1 − e−λu for u ≥ 0, and thus f(u) = F 0 (u) = λ e−λu .
Dung Nguyen
Probability and Statistics 23/44
The Exponential Distribution Exp(λ)
Continuous Distributions
N:
the number of flaws in u millimeters of wire.
X: the length from any starting point on the wire until a flaw
is detected
If the mean number of flaws is λ per millimeter then
(λu)0
= e−λu .
P(X > u) = P(N = 0) = e−λu
0!
Therefore, P(X ≤ u) = 1 − e−λu for u ≥ 0, and thus f(u) = F 0 (u) = λ e−λu .
X is called to be of an exponential distribution Exp(λ) if its pdf
satisfies
(
λ e−λx , x ≥ 0
f(x) =
.
0, x < 0
The cdf
(
1 − e−λx , x ≥ 0
F(x) =
0, x < 0
Dung Nguyen
Probability and Statistics 23/44
The Exponential Distribution Exp(λ)
Continuous Distributions
λ=1
λ = 3.5
λ=9
1.5
1
0.5
0
1
2
3
4
Figure: Pdf of Exp(λ)
Dung Nguyen
Probability and Statistics 24/44
The Exponential Distribution Exp(λ)
Continuous Distributions
Properties
If the random variable X has an exponential distribution with
parameter λ
Proposition (Basic properties)
1
E(X) =
1
λ
and
V(X) =
1
λ2
Poisson distribution: Mean and variance are same.
Exponential distribution: Mean and standard deviation are same.
2
For any s, t ≥ 0
P(X > s + t|X > s) = P(X > t)
Random sample (Exp(5)):
5.99 0.51 0.14 3.58 2.54
5.16 4.17 18.84 1.86 3.11
Then x = 4.59 and s = 5.3421.
Dung Nguyen
Random
10.70
18.68
Then x
sample (U(0, 20)):
2.52 13.24 11.43 5.68
14.23 19.97 11.43 12.97
= 12.085 and s = 5.2459.
Probability and Statistics 25/44
The Exponential Distribution Exp(λ)
Continuous Distributions
Example 14 - Computer Usage
In a large corporate computer network, user log-ons to the system
can be modeled as a Poisson process with a mean of 25 log-ons per
hour.
a
What is the probability that there are no log-ons in the next 6
minutes (0.1 hour)?
b
What is the probability that the time until the next log-on is
between 2 and 3 minutes (0.033 and 0.05 hours)?
c
What is the interval of time such that the probability that no
log-on occurs during the interval is 0.90?
d
What are the mean and standard deviation of the time until the
next log-on?
Dung Nguyen
Probability and Statistics 26/44
The Exponential Distribution Exp(λ)
Continuous Distributions
Example
15
Suppose that the time between detections of a particle with a
Geiger counter has an exponential distribution with the average
time 1.4 minutes. What is the probability that a particle is
detected in the next 30 seconds using the counter?
Dung Nguyen
Probability and Statistics 27/44
The Exponential Distribution Exp(λ)
Continuous Distributions
Example
15
Suppose that the time between detections of a particle with a
Geiger counter has an exponential distribution with the average
time 1.4 minutes. What is the probability that a particle is
detected in the next 30 seconds using the counter?
16
Suppose that a number of miles that a car can run before its
battery wears out is exponentially distributed with an average
value of 10000 miles. If a person desires to take a 5000-mile
trip, what is the probability that she will be able to complete
her trip without having to replace her car battery? What can
be said when the distribution is not exponential? (e.g.
U(0, 20000))
Dung Nguyen
Probability and Statistics 27/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Introduction
·10−2
0.15
8
0.1
6
4
0.05
2
0
0
0
5
10
15
20
25
30
0
10
Figure: B(50, 0.2) and Poisson(20)
Dung Nguyen
Probability and Statistics 28/44
20
30
40
The Normal Distributions N(m, σ 2 )
Continuous Distributions
The Normal Distributions N(m, σ 2 )
X is called to be of a normal distribution N(m, σ 2 ) if its pdf
satisfies
f(x) =
√1
σ 2π
−
e
(x−m)2
2σ 2
, x ∈ R,
where m = E(X) and σ 2 = V(X).
Dung Nguyen
Probability and Statistics 29/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
The Normal Distributions N(m, σ 2 )
X is called to be of a normal distribution N(m, σ 2 ) if its pdf
satisfies
f(x) =
√1
σ 2π
−
e
(x−m)2
2σ 2
, x ∈ R,
where m = E(X) and σ 2 = V(X).
We often standardize a normal distribution X ∼ N(m, σ 2 ) by
X −m
Y=
∼ N(0, 1)
σ
In this case, Y is called a random variable of the standard normal
distribution, or simply a standard score. Its pdf is
f(x) =
√1
2π
Dung Nguyen
2
− x2
e
,x ∈ R
Probability and Statistics 29/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
µ = 0, σ 2 = 1
µ = 0, σ 2 = 9
µ = −10, σ 2 = 9
0.4
0.3
0.2
0.1
0
−20
−10
0
10
Figure: Pdf of N(µ, σ 2 )
Dung Nguyen
Probability and Statistics 30/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
The cdf of X ∼ N(0, 1)
Z
x
Φ(x) =
f(u)du
−∞
satisfies
Φ(−x) = 1 − Φ(x)
and
Φ−1 (p) = −Φ−1 (1 − p),
for 0 < p < 1.
Denote zα as the solution to 1 − Φ(z) = α
this area = α
zα
x
zα is called the upper α critical point or the 100(1 − α)th
percentile.
Dung Nguyen
Probability and Statistics 31/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
17
Suppose that the current measurements in a strip of wire follow
a normal distribution with µ = 10 mA and σ = 2 mA.
Dung Nguyen
Probability and Statistics 32/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
17
Suppose that the current measurements in a strip of wire follow
a normal distribution with µ = 10 mA and σ = 2 mA.
a
What is the probability that the current measurement is between
9 mA and 11 mA?
Dung Nguyen
Probability and Statistics 32/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
17
Suppose that the current measurements in a strip of wire follow
a normal distribution with µ = 10 mA and σ = 2 mA.
a
b
What is the probability that the current measurement is between
9 mA and 11 mA?
Determine the value for which the probability that a current
measurement falls below this value is 0.98.
Dung Nguyen
Probability and Statistics 32/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
17
Suppose that the current measurements in a strip of wire follow
a normal distribution with µ = 10 mA and σ = 2 mA.
a
b
18
What is the probability that the current measurement is between
9 mA and 11 mA?
Determine the value for which the probability that a current
measurement falls below this value is 0.98.
Consider three independent memory chips. Suppose that the
lifetime of each memory chip has normal distribution with mean
300 hours and standard deviation 10 hours. Compute the
probability that at least one of three chips lasts at least 290
hours.
Dung Nguyen
Probability and Statistics 32/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Properties
Proposition (Basic properties)
Suppose that X ∼ N(m, σ 2 ).
1
E(X) = m and V(X) = σ 2
2
If Y = aX + b, a 6= 0 then
Y ∼ N(am + b, a2 σ 2 ).
3
If Xi ∼ N(mi , σi2 ) and are independent then
n
X
Xi ∼ N(
i=1
Dung Nguyen
n
X
i=1
mi ,
n
X
!
σi2
.
i=1
Probability and Statistics 33/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
19
Let X ∼ N(5, 9).
What is the distribution of Y = 2X − 6?
Dung Nguyen
Probability and Statistics 34/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
19
Let X ∼ N(5, 9).
What is the distribution of Y = 2X − 6?
20
Let X1 ∼ N(2, 4) and X2 ∼ N(−3, 5) be independent. Determine the
distributions of the following random variables
Dung Nguyen
Probability and Statistics 34/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
19
Let X ∼ N(5, 9).
20
Let X1 ∼ N(2, 4) and X2 ∼ N(−3, 5) be independent. Determine the
distributions of the following random variables
a
What is the distribution of Y = 2X − 6?
X = X1 + X2 .
Dung Nguyen
Probability and Statistics 34/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
19
Let X ∼ N(5, 9).
20
Let X1 ∼ N(2, 4) and X2 ∼ N(−3, 5) be independent. Determine the
distributions of the following random variables
a
X = X1 + X2 .
What is the distribution of Y = 2X − 6?
b
Y = X1 − X2 .
Dung Nguyen
Probability and Statistics 34/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
19
Let X ∼ N(5, 9).
20
Let X1 ∼ N(2, 4) and X2 ∼ N(−3, 5) be independent. Determine the
distributions of the following random variables
a
X = X1 + X2 .
What is the distribution of Y = 2X − 6?
b
Y = X1 − X2 .
Dung Nguyen
c
Z = 3X1 + 4X2 .
Probability and Statistics 34/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
19
Let X ∼ N(5, 9).
20
Let X1 ∼ N(2, 4) and X2 ∼ N(−3, 5) be independent. Determine the
distributions of the following random variables
a
21
X = X1 + X2 .
What is the distribution of Y = 2X − 6?
b
Y = X1 − X2 .
c
Z = 3X1 + 4X2 .
Data from the National Oceanic and Atmospheric Administration
indicate that the yearly precipitation in Los Angeles is a
normal random variable with a mean of 12.08 inches and a
standard deviation of 3.1 inches. Assume that the
precipitation totals for the next 2 years are independent.
Dung Nguyen
Probability and Statistics 34/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
19
Let X ∼ N(5, 9).
20
Let X1 ∼ N(2, 4) and X2 ∼ N(−3, 5) be independent. Determine the
distributions of the following random variables
a
21
X = X1 + X2 .
What is the distribution of Y = 2X − 6?
b
Y = X1 − X2 .
c
Z = 3X1 + 4X2 .
Data from the National Oceanic and Atmospheric Administration
indicate that the yearly precipitation in Los Angeles is a
normal random variable with a mean of 12.08 inches and a
standard deviation of 3.1 inches. Assume that the
precipitation totals for the next 2 years are independent.
a
Find the probability that the total precipitation during the next
2 years will exceed 25 inches.
Dung Nguyen
Probability and Statistics 34/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
19
Let X ∼ N(5, 9).
20
Let X1 ∼ N(2, 4) and X2 ∼ N(−3, 5) be independent. Determine the
distributions of the following random variables
a
21
X = X1 + X2 .
What is the distribution of Y = 2X − 6?
b
Y = X1 − X2 .
c
Z = 3X1 + 4X2 .
Data from the National Oceanic and Atmospheric Administration
indicate that the yearly precipitation in Los Angeles is a
normal random variable with a mean of 12.08 inches and a
standard deviation of 3.1 inches. Assume that the
precipitation totals for the next 2 years are independent.
a
b
Find the probability that the total precipitation during the next
2 years will exceed 25 inches.
Find the probability that next year’s precipitation will exceed
that of the following year by more than 3 inches.
Dung Nguyen
Probability and Statistics 34/44
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Revisit sums random variables
1
If Xi ∼ Poisson(λi ) and are independent then
!
n
n
X
X
Xi ∼ Poisson
λi .
i=1
2
If Xi ∼
N(mi , σi2 )
i=1
and are independent then
n
X
Xi ∼ N
i=1
Dung Nguyen
n
X
i=1
mi ,
n
X
!
σi2
.
i=1
Probability and Statistics 35/44
Continuous Distributions
Central Limit Theorem
Normal Approximations
The binomial and Poisson distributions become more bell-shaped and
symmetric as their mean values increase.
·10−2
0.15
8
0.1
6
4
0.05
2
0
0
0
5
10
15
20
25
30
0
Nguyen
and Statistics
Figure:Dung
B(50,
0.2) Probability
and Poisson(20)
10
36/44
20
30
40
Continuous Distributions
Central Limit Theorem
Assume X1 , X2 , . . . are independent such that
Denote Sn =
n
X
i=1
Xi
E(Xk ) = m and V(Xk ) = σ 2 .
n
1X
and X n =
Xi . Then
n i=1
E(Sn ) = nm,
V(Sn ) = nσ 2 ,
√
SD(Sn ) = σ n,
V(X n ) = σ 2 /n,
√
SD(X) = σ/ n.
and
E(X n ) = m,
Dung Nguyen
Probability and Statistics 37/44
Continuous Distributions
Central Limit Theorem
Proposition
Assume X1 , X2 , . . . ∼ N(m, σ 2 ) and are independent such that σ 2 < ∞.
n
n
X
1X
Xi . Then
Denote Sn =
Xi and X n =
n i=1
i=1
Sn −E(Sn )
SD(Sn )
and
X n −E(X n )
SD(X n )
Sn − nm
= √ ∼ N(0, 1),
σ n
Xn − m
= √ ∼ N(0, 1).
σ/ n
Dung Nguyen
Probability and Statistics 38/44
Continuous Distributions
Central Limit Theorem
The Central Limit Theorem
Proposition (Central Limit Theorem)
Assume X1 , X2 , . . . are i.i.d. (independent identically distributed)
n
X
such that E(X) = m and V(X) = σ 2 < ∞. Denote Sn =
Xi and
i=1
n
1X
Xn =
Xi .
n i=1
Then
Sn −E(Sn )
SD(Sn )
Sn − nm
= √ ' N(0, 1) as n → ∞.
σ n
and
X n −E(X n )
SD(X n )
Xn − m
= √ ' N(0, 1) as n → ∞.
σ/ n
Dung Nguyen
Probability and Statistics 39/44
Continuous Distributions
Central Limit Theorem
Example
22
A producer of cigarettes claims that the mean nicotine content
in its cigarettes is 2.4 milligrams with a standard deviation
of 0.2 milligrams. Assuming these figures are correct,
approximate the probability that the sample mean of 100
randomly chosen cigarettes is
Dung Nguyen
Probability and Statistics 40/44
Continuous Distributions
Central Limit Theorem
Example
22
A producer of cigarettes claims that the mean nicotine content
in its cigarettes is 2.4 milligrams with a standard deviation
of 0.2 milligrams. Assuming these figures are correct,
approximate the probability that the sample mean of 100
randomly chosen cigarettes is
a
Greater than 2.5 milligrams.
Dung Nguyen
Probability and Statistics 40/44
Continuous Distributions
Central Limit Theorem
Example
22
A producer of cigarettes claims that the mean nicotine content
in its cigarettes is 2.4 milligrams with a standard deviation
of 0.2 milligrams. Assuming these figures are correct,
approximate the probability that the sample mean of 100
randomly chosen cigarettes is
a
b
Greater than 2.5 milligrams.
Less than 2.25 milligrams.
Dung Nguyen
Probability and Statistics 40/44
Continuous Distributions
Central Limit Theorem
Normal Approximation to the Binomial Distribution
If X is a binomial random variable with parameters n and p then
X − E(X) X − np
=√
' N(0, 1).
SD(X)
npq
The approximate is good if np > 5 and n(1 − p) > 5
Continuity correction
k + 0.5 − np
P(X ≤ k) = P(X < k + 0.5) ≈ P Z <
√
npq
and
k − 0.5 − np
P(X ≥ k) = P(X > k − 0.5) ≈ P Z >
√
npq
Dung Nguyen
Probability and Statistics 41/44
Continuous Distributions
Central Limit Theorem
Example
Suppose only 75% of all drivers in a certain state regularly wear a
seat belt. A random sample of 500 drivers is selected. What is
the probability that
a
Between 360 and 400 (inclusive) of the drivers in the sample
regularly wear a seat belt?
b
Fewer than 400 of those in the sample regularly wear a seat
belt?
Dung Nguyen
Probability and Statistics 42/44
Continuous Distributions
Central Limit Theorem
Normal Approximation to the Poisson Distributions
If X is a Poisson random variable with E(X) = λ and V(X) = λ,
X − E(X) X − λ
= √
' N(0, 1)
SD(X)
λ
Continuity correction
The approximation is good for λ ≥ 5.
Dung Nguyen
Probability and Statistics 43/44
Continuous Distributions
Central Limit Theorem
Example
Assume that the number of asbestos particles in a square meter of
dust on a surface follows a Poisson distribution with a mean of
1000. If a square meter of dust is analyzed, what is the
probability that 950 or fewer particles are found?
Dung Nguyen
Probability and Statistics 44/44