HCMC University of Technology
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)
Probability and Statistics
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Some Special Distributions
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The Binomial Distributions B(n, p)
Discrete Distributions
Definition
Bernoulli trial B(p):
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).
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 Statistics3 / 32
The Binomial Distributions B(n, p)
Discrete Distributions
The Binomial Distributions B(n, p)
Probability and Statistics2 / 32
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Probability and Statistics4 / 32
The Binomial Distributions B(n, p)
Discrete Distributions
The Binomial Distributions B(n, p)
Discrete Distributions
Proposition
n = 30, p = 0.5
n = 30, p = 0.2
n = 8, p = 0.5
n = 8, p = 0.2
0.3
Let X ∼ B(n, p).
1
0.15
Then
X takes values in Ω = {0, 1, . . . , n} such that
f(k) = P(X = k) = Ckn pk q n−k .
0.2
0.1
0.1
0.05
0
0
k = 0:
k = n:
k = 1:
k = 2:
2
0
2
4
6
8
0
5
10
15
20
25
30
3
···
···
···
···
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 .
(
1,
where Zi =
0,
if the i-th trial is successful
.
otherwise
E(X) = np
V(X) = npq.
and
Probability and Statistics5 / 32
Dung Nguyen
Probability and Statistics6 / 32
The Binomial Distributions B(n, p)
Discrete Distributions
Example
2
q
p
q
q
X is a sum of n independent Bernoulli random variables.
The Binomial Distributions B(n, p)
Discrete Distributions
q
p
q
p
X = Z1 + Z2 + · · · + Zn ,
Figure: Pmf of B(n, p)
Dung Nguyen
q
p
p
p
Example
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.
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
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Probability and Statistics7 / 32
Find the probability that, in the next 18 samples, exactly 2
contain the pollutant.
Dung Nguyen
Probability and Statistics7 / 32
The Binomial Distributions B(n, p)
Discrete Distributions
Example
2
Example
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
2
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.
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
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c
3
Example
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.
b
Probability and Statistics7 / 32
The Binomial Distributions B(n, p)
Discrete Distributions
Example
a
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.
Probability and Statistics7 / 32
The Binomial Distributions B(n, p)
Discrete Distributions
2
The Binomial Distributions B(n, p)
Discrete Distributions
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.
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.
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 Statistics7 / 32
Dung Nguyen
Probability and Statistics8 / 32
The Binomial Distributions B(n, p)
Discrete Distributions
Example
4
5
Applications of Poisson Distributions
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.
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?
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Known:
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
The Poisson Distributions
Unknown:
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.
Probability and Statistics8 / 32
The Poisson Distributions Poisson(λ)
Discrete Distributions
Poisson r.v.
The Poisson Distributions Poisson(λ)
Discrete Distributions
Probability and Statistics9 / 32
The Poisson Distributions Poisson(λ)
Discrete Distributions
The mean and the variance of a Poisson distribution are the same.
= the count of events that occur within an interval.
E(X) = λ
V(X) = λ.
and
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!
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
Poisson distribution:
Ω = {0, 1, 2, . . .}
and
f(k) = P(X = k) = e−λ ·
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λk
,
k!
x ∈ Ω.
Probability and Statistics10 / 32
Xi ∼ Poisson
i=1
n
X
!
λi
.
i=1
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Probability and Statistics11 / 32
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
·10−2
λ = 0.5
0.6
6
λ = 20
λ=5
8
0.15
6
0.4
The Poisson Distributions Poisson(λ)
Discrete Distributions
0.1
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?
4
0.2
0.05
0
2
0
0
0
2
4
6
8
0
5
10
15
0
10
20
30
40
Figure: Pmf of Poisson(λ)
Dung Nguyen
Probability and Statistics12 / 32
Dung Nguyen
The Poisson Distributions Poisson(λ)
Discrete Distributions
Probability and Statistics13 / 32
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
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?
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
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
Dung Nguyen
Probability and Statistics13 / 32
exactly 2 flaws in 1 mm of wire.
Dung Nguyen
Probability and Statistics13 / 32
The Poisson Distributions Poisson(λ)
Discrete Distributions
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
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?
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
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
a
b
c
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Probability and Statistics13 / 32
The Poisson Distributions Poisson(λ)
Discrete Distributions
Probability and Statistics13 / 32
The Poisson Distributions Poisson(λ)
Discrete Distributions
Approximation property
Example
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)
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.
Poisson(4)
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?
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.
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10
15
0
5
10
15
B(n, p) with np = λ
Probability and Statistics14 / 32
Dung Nguyen
Probability and Statistics15 / 32
The Poisson Distributions Poisson(λ)
Discrete Distributions
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
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?
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
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
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Probability and Statistics15 / 32
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The Poisson Distributions Poisson(λ)
Discrete Distributions
Be at most 10?
Probability and Statistics15 / 32
The Poisson Distributions Poisson(λ)
Discrete Distributions
Example
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?
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
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?
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 Statistics15 / 32
Dung Nguyen
Probability and Statistics15 / 32
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
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
0.4


0,

x − a
,
F(x) =

b−a


1,
x<a
U(0, 2.5)
U(0, 5)
U(0, 10)
U(−2, 2)
0.3
x ∈ [a, b) .
0.2
x≥b
0.1
0
−2
0
2
4
6
8
10
Figure: Pdf of U(a, b)
Proposition (Properties)
a+b
1 E(X) =
2
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Probability and Statistics16 / 32
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The Continuous Uniform Distributions U(a, b)
Continuous Distributions
V(X) =
(b − a)2
12
Probability and Statistics17 / 32
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
Example
11
2
Example
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
11
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
a
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Probability and Statistics18 / 32
What is the probability that the current is between 4.95 mA and
5.0 mA?
Dung Nguyen
Probability and Statistics18 / 32
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
Example
11
Example
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
a
b
11
What is the probability that the current is between 4.95 mA and
5.0 mA?
Calculate its mean and variance.
Dung Nguyen
b
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
b
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 Statistics18 / 32
The Exponential Distribution Exp(λ)
Continuous Distributions
The Exponential Distribution Exp(λ)
Example
a
What is the probability that the current is between 4.95 mA and
5.0 mA?
Calculate its mean and variance.
Probability and Statistics18 / 32
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
Let X be a measurement of current, which is a variable
following a continuous uniform distribution on [4.9, 5.1].
a
12
11
The Continuous Uniform Distributions U(a, b)
Continuous Distributions
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 Statistics18 / 32
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 Statistics19 / 32
The Exponential Distribution Exp(λ)
Continuous Distributions
N:
The Exponential Distribution Exp(λ)
Continuous Distributions
the number of flaws in u millimeters of wire.
X: the length from any starting point on the wire until a flaw
is detected
λ=1
λ = 3.5
λ=9
1.5
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 .
1
0.5
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
0
1
4
(
1 − e−λx , x ≥ 0
F(x) =
0, x < 0
Dung Nguyen
Probability and Statistics20 / 32
Dung Nguyen
The Exponential Distribution Exp(λ)
Continuous Distributions
Proposition (Basic properties)
1
λ
and
The Exponential Distribution Exp(λ)
Example 14 - Computer Usage
If the random variable X has an exponential distribution with
parameter λ
E(X) =
Probability and Statistics21 / 32
Continuous Distributions
Properties
V(X) =
1
λ2
Poisson distribution: Mean and variance are same.
Exponential distribution: Mean and standard deviation are same.
2
3
Figure: Pdf of Exp(λ)
The cdf
1
2
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 the standard deviation of the time until
the next log-on?
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 Statistics22 / 32
Dung Nguyen
Probability and Statistics23 / 32
The Exponential Distribution Exp(λ)
Continuous Distributions
Example
15
The Exponential Distribution Exp(λ)
Continuous Distributions
Example
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
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 Statistics24 / 32
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Probability and Statistics24 / 32
The Normal Distributions N(m, σ 2 )
Continuous Distributions
The Normal Distributions N(m, σ 2 )
Introduction
X is called to be of a normal distribution N(m, σ 2 ) if its pdf
satisfies
·10−2
0.15
8
0.1
f(x) =
6
√1
σ 2π
−
e
(x−m)2
2σ 2
, x ∈ R,
where m = E(X) and σ 2 = V(X).
4
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
0.05
2
0
0
0
5
10
15
20
25
30
0
10
20
Figure: B(50, 0.2) and Poisson(20)
Dung Nguyen
Probability and Statistics25 / 32
30
40
f(x) =
√1
2π
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2
− x2
e
,x ∈ R
Probability and Statistics26 / 32
The Normal Distributions N(m, σ 2 )
Continuous Distributions
The Normal Distributions N(m, σ 2 )
Continuous Distributions
The cdf of X ∼ N(0, 1)
µ = 0, σ 2 = 1
µ = 0, σ 2 = 9
µ = −10, σ 2 = 9
0.4
0.3
Z
satisfies
and
Φ−1 (p) = −Φ−1 (1 − p),
this area = α
−20
−10
0
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zα
10
Figure: Pdf of N(µ, σ 2 )
x
zα is called the upper α critical point or the 100(1 − α)th
percentile.
Probability and Statistics27 / 32
Dung Nguyen
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Probability and Statistics28 / 32
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
17
for 0 < p < 1.
Denote zα as the solution to 1 − Φ(z) = α
0.1
0
f(u)du
−∞
Φ(−x) = 1 − Φ(x)
0.2
x
Φ(x) =
Example
Suppose that the current measurements in a strip of wire follow
a normal distribution with µ = 10 mA and σ = 2 mA.
17
Suppose that the current measurements in a strip of wire follow
a normal distribution with µ = 10 mA and σ = 2 mA.
a
Dung Nguyen
Probability and Statistics29 / 32
What is the probability that the current measurement is between
9 mA and 11 mA?
Dung Nguyen
Probability and Statistics29 / 32
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
Example
Suppose that the current measurements in a strip of wire follow
a normal distribution with µ = 10 mA and σ = 2 mA.
17
a
b
The Normal Distributions N(m, σ 2 )
Continuous Distributions
17
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.
a
b
18
Dung Nguyen
Suppose that the current measurements in a strip of wire follow
a normal distribution with µ = 10 mA and σ = 2 mA.
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.
Probability and Statistics29 / 32
Dung Nguyen
The Normal Distributions N(m, σ 2 )
Continuous Distributions
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Properties
Probability and Statistics29 / 32
Example
Proposition (Basic properties)
19
Let X ∼ N(5, 9).
What is the distribution of Y = 2X − 6?
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 Statistics30 / 32
Dung Nguyen
Probability and Statistics31 / 32
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
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
What is the distribution of Y = 2X − 6?
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
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X = X1 + X2 .
Dung Nguyen
The Normal Distributions N(m, σ 2 )
Probability and Statistics31 / 32
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
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
X = X1 + X2 .
What is the distribution of Y = 2X − 6?
Probability and Statistics31 / 32
Continuous Distributions
a
The Normal Distributions N(m, σ 2 )
Continuous Distributions
What is the distribution of Y = 2X − 6?
b
Y = X1 − X2 .
Dung Nguyen
Probability and Statistics31 / 32
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 Statistics31 / 32
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Example
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
What is the distribution of Y = 2X − 6?
X = X1 + X2 .
b
Y = X1 − X2 .
c
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
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
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X = X1 + X2 .
Dung Nguyen
The Normal Distributions N(m, σ 2 )
Example
Y = X1 − X2 .
b
Z = 3X1 + 4X2 .
Probability and Statistics31 / 32
The Normal Distributions N(m, σ 2 )
Continuous Distributions
Revisit sums random variables
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
What is the distribution of Y = 2X − 6?
b
Y = X1 − X2 .
c
Z = 3X1 + 4X2 .
1
If Xi ∼ Poisson(λi ) and are independent then
!
n
n
X
X
Xi ∼ Poisson
λi .
2
If Xi ∼ N(mi , σi2 ) and are independent then
i=1
n
X
Xi ∼ N
i=1
21
c
Find the probability that the total precipitation during the next
2 years will exceed 25 inches.
Probability and Statistics31 / 32
Continuous Distributions
X = X1 + X2 .
What is the distribution of Y = 2X − 6?
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.
21
a
a
The Normal Distributions N(m, σ 2 )
Continuous Distributions
i=1
n
X
i=1
mi ,
n
X
!
σi2
.
i=1
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 Statistics31 / 32
Dung Nguyen
Probability and Statistics32 / 32
Continuous Distributions
Central Limit Theorem
Continuous Distributions
Normal Approximations
Central Limit Theorem
Assume X1 , X2 , . . . are independent such that
The binomial and Poisson distributions become more bell-shaped and
symmetric as their mean values increase.
·10−2
0.15
n
X
Xi
i=1
E(Sn ) = nm,
8
0.1
Denote Sn =
E(Xk ) = m and V(Xk ) = σ 2 .
n
1X
and X n =
Xi . Then
n i=1
V(Sn ) = nσ 2 ,
√
SD(Sn ) = σ n,
V(X n ) = σ 2 /n,
√
SD(X) = σ/ n.
and
E(X n ) = m,
6
4
0.05
2
0
0
0
5
10
15
20
25
30
0
10
20
30
40
Nguyen
and Statistics33 / 32
Figure:Dung
B(50,
0.2) Probability
and Poisson(20)
Continuous Distributions
Dung Nguyen
Central Limit Theorem
Continuous Distributions
Probability and Statistics34 / 32
Central Limit Theorem
The Central Limit Theorem
Proposition
Assume X1 , X2 , . . . ∼ N(m, σ 2 ) and are independent such that σ 2 < ∞.
n
n
X
1X
Denote Sn =
Xi and X n =
Xi . Then
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
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 )
Probability and Statistics35 / 32
Sn − nm
√ ' N(0, 1) as n → ∞.
σ n
and
X n −E(X n )
SD(X n )
Dung Nguyen
=
=
Xn − m
√ ' N(0, 1) as n → ∞.
σ/ n
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Probability and Statistics36 / 32
Continuous Distributions
Central Limit Theorem
Continuous Distributions
Example
22
Example
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 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
22
a
Dung Nguyen
Continuous Distributions
Dung Nguyen
Central Limit Theorem
Continuous Distributions
Probability and Statistics37 / 32
Central Limit Theorem
Normal Approximation to the Binomial Distribution
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.
Probability and Statistics37 / 32
Example
22
Central Limit Theorem
Greater than 2.5 milligrams.
Less than 2.25 milligrams.
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
Dung Nguyen
Probability and Statistics37 / 32
k − 0.5 − np
P(X ≥ k) = P(X > k − 0.5) ≈ P Z >
√
npq
Dung Nguyen
Probability and Statistics38 / 32
Continuous Distributions
Central Limit Theorem
Example
Continuous Distributions
Central Limit Theorem
Normal Approximation to the Poisson Distributions
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
Continuous Distributions
Probability and Statistics39 / 32
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 Statistics41 / 32
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 Statistics40 / 32