Random Variables and
Probability Distributions
KANCHALA SUDTACHAT
Content
1.
Random Variables
9.
Normal Distribution
2.
Probability
10. t- Distribution
3.
Discrete Random Variables
11. Exponential Distribution
4.
Discrete Uniform Distribution
5.
Binomial Distribution
12. Normal Approximation to the
Binomial and Poisson
Distributions
6.
Poisson Distribution
13. Normal Probability Plots
7.
Continuous Random Variables
8.
Continuous Uniform
Distribution
Random Experiment
Random Variables
A measurement is usually denoted by a variable such as X.
A variable whose measured value can change. (random variable)
Discrete Random Variable – a finite set of real numbers
Continuous Random Variable – an interval of real numbers
Probability
A random variable is used to describe a measurement.
Probability is used to quantify the likelihood, that a measurement falls
within some set of values.
“The chance that X, the length of a manufactured part, is
between 10.8 and 11.2 millimeters is 25%”
Probability
 Degree of belief
 A relative frequency (or proportion) of repeated replicates that fall in the
interval will be percent, uses a long run proportion.
Probability Properties
Given a set E, the set of real number is denoted as R.
Each element is in one and only one of the sets E1, E2,…,Ek.
Applying Probability Properties
Ex: X denotes the life in hours of standard fluorescent tubes
P(X ≤ 5000) = 0.1, P(5000<X ≤ 6000) = 0.3, P(X > 8000) = 0.4
Discrete Random Variables
Example 3-19 A voice communication network for a business
contains 48 external lines. At a particular time, the system is observed
and some of the lines are being used.
Example 3-20 The analysis of the surface of a semiconductor
wafer records the number of particles of contamination that exceed a
certain size. Define the random variable X to equal the number of
particles of contamination.
Probability Mass Function (pmf)
The probability distribution of a random variable X is a description of
the probabilities associated with the possible values of X.
It is convenient to express the probability in terms of a formula.
Example 3-21 There is a chance that a bit transmitted through a
digital transmission channel is received in error. Let X equal the number
of bits in error in the next 4 bits transmitted. The possible value for X are
{0, 1, 2, 3, 4}
Probability Mass Function (pmf)
Cumulative Distribution Function (cdf)
Cumulative Distribution
Function (cdf)
Example 3-22 In previous example, the probability mass function for X
P(X=0) = 0.6561
P(X=1) = 0.2916
P(X=3) = 0.0036
P(X=4) = 0.0001
P(X=2) = 0.0486
Mean and Variance
Example 3-23 For the number of bits in error in the previous
example. Determining µ and σ2
Mean and Variance
Example 3-24 Two new product designs are to be compared on
the basis for revenue potential. Marketing feels that the revenue from
design A can be predicted quite accurately to be $3 million. The revenue
potential of design B is more difficult to assess. Marketing concludes that
there is a probability of 0.3 that the revenue from design B will be $7
million, but there is a 0.7 probability that the revenue will be only $2
million. Which design would you choose?
Discrete Uniform Distribution
Example The first digit of a part’s serial number is equally likely to
be any one of the digits 0 through 9. If one part is selected from a large
batch and X is the first digit of the serial number.
Discrete Uniform Distribution
Example As in previous example 3-19, let the random variable X
denote the number of the 48 voice lines that are in use at a particular
time. Assume that X is a discrete uniform random variable with a range of
0 to 48. Determining µ and σ2
Binomial Distribution
Each of these random experiments can be thought of as consisting of a
series of repeated, random trials.
The random variable in each case is a count of the number of trials that
meet a specified criterion.
Binomial Distribution
Binomial Distribution
Example 3-25 In Example 3-21, assume that the chance that a
bit transmitted through a digital transmission channel is received in
error is 0.1. Also assume that the transmission trials are independent.
Let X = the number of bits in error in the next 4 bits transmitted.
Determine P (X=2)
Binomial Distribution
Example 3-27 Each sample of water has a 10% chance of
containing high levels of organic solids. Assume that the samples are
independent with regard to the presence of the solids. Determine the
probability that in the next 18 samples, exactly 2 contain high solids.
Example 3-28 For the number of transmitted bits received in
error in Example 3-21, n = 4 and p = 0.1
Poisson Process
Events occur randomly in an interval time.
The number of events over an interval time is a discrete random
variable.
Poisson Process
Example 3-30 Flaws occur at random along the length of the
thin copper wire. Let X denote the random variable that counts the
number of flaws in a length of L millimeters of wire and suppose that
the average number of flaws in L millimeters is λ.
Poisson Process
In general, consider an interval T of real number partitioned into
subintervals of small length Δt and assume that as Δt tends to zero,
1) The probability of more than one event in a subinterval tends to zero,
2) The probability of one event in a subinterval tends to λ Δt /T
3) The event in each subinterval is independent of other subintervals
A random experiment with these properties is called a Poisson Process.
Poisson Process
Poisson Process
Example 3-31 For the case of the thin copper wire, suppose that the
number of flaws follows a Poisson distribution with a mean of 2.3 flaws
per millimeter. Determine the probability of exactly 2 flaws in 1
millimeter of wire.
Poisson Process
Example 3-32 Contamination is a problem in the manufacture
of optical storage disks. The number of particles of contamination that
occur on an optical disk has a Poisson distribution, and the average
number of particles per centimeter squared of media surface is 0.1. The
area of a disk under study is 100 squared centimeters. Determine the
probability that 12 particles occur in the area of a disk under study.
Continuous Random Variables
Probability density Function f(x) are used to describe physical
systems.
Considering the density of a loading on a long, thin beam. For any point
x along the beam, the density can be described by a functions (in
grams/cm).
Probability density function
A pdf is zero for x values that cannot occur
f(x) is used to calculate an area the represents the probability that X
assumes a value in [a, b].
Probability density function
Example 3-2 Let the continuous random variable X denote the
current measured in a thin copper wire in mill-amperes. Assume that
the range of X is [0, 20 mA], and assume that the probability density
function of X is f(x)=0.05 for 0 ≤ x ≤ 20. What is the probability that a
current measurement is less than 10 mill-amperes?
Probability density function
Example 3-3 Let the continuous random variable X denote the distance
in micrometers from the start of a track on a magnetic disk until the first
flaw. Historical data show that the distribution of X can be modeled by a
pdf
For what proportion of disks is the distance
to the first flaw greater than 1000 micrometers?
Cumulative Distribution Function
Cumulative Distribution Function
Example 3-4 Consider the distance to flaws in Example 3-3 with pdf
f(x) = 1/2000 exp(-x/2000)
For x ≥ 0. Determine the cdf.
Mean and Variance
Mean and Variance
Example 3-5
For the copper current measurement in Example 3-2,
the mean of X is
Example 3-6
of X is
For the distance to a flaw in Example 3-3, the mean
Normal Distribution
Histograms have characteristic shape as bell shapes.
The random variable that equals the average result over the replicates
tends to have a normal distribution as the number of replicates
becomes large.
Normal Distribution
Normal Distribution
Example 3-7 Assume that the current measurements in a
strip of wire follow a normal distribution with a mean of 10
mill-amperes and a variance of 4 mill-amperes2. What is the
probability that a measurement exceeds 13 mill-amperes?
Normal Distribution
Standard normal random variable
Using standard normal random
variable
Example 3-8 Assume that Z is a standard normal random variable.
Appendix A Table I provides probabilities of the form P(Z ≤ z). The use of
Table I to fine P(Z ≤ 1.5).
Normal Distribution
39
Example 3-9
1) P(Z > 1.26)
2) P(Z < -0.86)
3) P(Z > -1.37)
4) P(Z < 1.37)
5) P(-1.25 < Z < 0.37)
6) P(Z ≤ -4.6)
7) Fine the value z such that P(Z > z) = 0.05
8) Find the value of z such that P(-z < Z < z)=0.99
Example 3-10
Suppose the current measurements in a strip of wire follow a normal
distribution with a mean of 10 mill-amperes and a variance of 4 millamperes2. What is the probability that a measurement exceeds 13 millamperes?
Standardizing
Example 3-11
Continuing the previous example, what is the probability that a current
measurement is between 9 and 11 mill-amperes?
Example 3-12
In the transmission of a digital signal, assume that the background noise
follows a normal distribution with a mean of 0 volt and standard
deviation of 0.45 volt. If the system assumes that a digital 1 has been
transmitted when the voltage exceeds 0.9, what is the probability of
detecting a digital 1 when none was sent?
Example 3-13
The diameter of a shaft in storage drive is normally distributed with
mean 0.2508 inch and standard deviation 0.0005 inch. The
specifications on the shaft are 0.2500 ± 0.0015 inch. What proportion of
shafts conforms to specifications?
t-Distribution
When  is unknown
Small sample size
Degree of freedom (k) = n-1
Significant level = 
t, k
46
t-Distribution
47
Exponential Distribution
 Let the random variable X denote the length from any starting point on
the wire until a flaw is detected.
Let the random variable N denote the number of flaws in x mill-meters
of wire. Assume that the mean number of flaws is λ per mill-meters.
Exponential Distribution
Example 3-33
In a large corporate computer network, use log-on to the system can be
modeled as a Poisson process with a mean of 25 log-on per hour. What
is the probability that there are no log-on in an interval of 6 minutes?
Normal Approximation to the
Binomial
A random variable can be approximated with a normal random variable
when n is large.
Normal Approximation to the
Binomial
Example 3-34 In a digital communication channel, assume that the
number of bits received in error can be modeled by a binomial random
variable, and assume that the probability that a bit is received in error
in 1x10-5. If 16 million bits are transmitted, what is the probability that
more than 150 errors occur?
Example 3-35
Again consider the transmission of bits in the previous example. To
judge how well the normal approximation works, assume that only n=50
bits are to be transmitted and that the probability of an error is p=0.1.
The exact probability that 2 or fewer errors occur is
Normal Approximation to the
Poisson
Recall that the Poisson distribution was developed as the limit of a
binomial distribution as the number of trials increased to infinity.
Consequently, the normal distribution can also be used to approximate
probabilities of a Poisson random variable.
The approximation is good for λ > 5.
Normal Probability Plots
How do we know whether a normal distribution is a reasonable model
for data?
Probability plotting is a graphical method for visual examination of the
data.
Probability plotting typically uses special graph paper, known as
probability paper, that has been designed for the hypothesized
distribution.
The sample are first ranked for smallest to largest x1, x2 ,…, xn
The ordered observations x(j) are then plotted against their observed
cumulative frequency (j - 0.5)/n …… on the appropriate probability plot.
Normal Probability Plots
Example 3-18 Ten observations on the effective service life in
minutes of batteries used in a portable personal computer are as
follows: 176, 191, 214, 220, 205, 192, 201, 190, 183, 185. We
hypothesize that battery life is adequately modeled by a normal
distribution. To use probability plotting to investigate this hypothesis
Normal Probability Plots
 A normal probability plot can also be constructed on ordinary graph
paper by plotting the standardized normal scores zj against x(j)
(j – 0.5)/n = P(Z ≤ zj) = ɸ(zj)
Random Samples, Statistics and the
Central Limit Theorem
58
Random Samples, Statistics
and the Central Limit Theorem
Example 3-46 The average tensile strength of eight rubber O-rings
was 7274 kPa. Two obvious questions are the following: What can we
conclude about the average tensile strength of future O-rings? How
wrong might we be if we concluded that the average tensile strength of
this future population of O-rings is 7274?
Random Samples, Statistics and the
Central Limit Theorem
x1  x2  ...  xn
X
n
E( X )  
V (X ) 
2
n
60
Random Samples, Statistics
and the Central Limit Theorem
Example 3-47 Soft-drink cans are filled by an automated filling
machine. The mean fill volume is 358 ml, and the standard deviation is
1.5 ml. Assume that the fill volumes of the cans are independent,
normal random variables. What is the probability that the average
volume of 10 cans selected from this process is less than 355 ml?
Random Samples, Statistics and the
Central Limit Theorem
62
Random Samples, Statistics and the
Central Limit Theorem
63
Questions?