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Chapter Four
Continuous Random
Variables &
Probability
Distributions
Continuous Probability Distribution
(pdf) Definition:
b
P(a  X b) =  f(x)dx
a
For continuous RV X & a  b.
Two Conditions for a pdf:
1) f(x)  0 for all x
+
2)  f(x)dx = 1
-
Cumulative Distribution Function
Definition:
a
F(a) = P(X  a) =  f(x)dx
-
For continuous RV X.
PDF Example
Given pdf:
f(x) = cx2, for 0  x  2
f(x) = 0, elsewhere.
Find the value of c for which f(x)
will be a valid pdf.
Find the probability that 1  x  2.
Cumulative Distribution Function
Plot the CDF for RV X:
F(x) = 0
F(x) = x
F(x) = 1
for x < 0
for 0  x  1
for x > 1.
Expected Values for Continuous RV
+
E(X) = x =  x f(x)dx
-
Variance
+
V(X) = 2x =  (x-)2 f(x)dx=E[(X-)2]
-
Shortcut: V(X) = E(X2) – [E(X)]2
Expected Value of RV X Example
For a RV X, the pdf is given by:
f(x) = 0
for x < 50
f(x) = 1/20 for 50  x  70
f(x) = 0
for x > 70.
What is the expected value &
variance of RV X?
Expected Value & Variance Example
An electronic component is tested to
failure. Let continuous RV X measure the
time to failure for this component. The
pdf for this electronic component is
shown to be in hours by the following:
f(t) = 0
f(t) = e-t/1,000
1,000
for t < 0
for t  0.
What is the expected value & variance of
RV X?
Continuous Probability Distributions
Uniform
Normal
Gamma
Exponential
Chi-Squared
Weibull
Lognormal
Beta
Uniform Distribution
f(x; a, b) =
1
b-a
axb
Uniform pdf Example
A telephone call is equally likely
to occur any time between Noon &
1 PM. Let X be a RV that is 0 at
Noon & is the time of the call in
fractions of a hour past Noon for
any other outcome.
Find P(0  X  1.0).
Find P(X = 0.2).
Find P(X  0.5).
Find P(0.2 < X < 0.8).
Uniform Probability Distribution
Resistors are produced that have a
nominal value of 10 ohms & are +/10% resistors where any possible
value of resistance is equally
likely.
Find the pdf & cdf of the RV X,
which represents resistance.
Find the probability that a resistor,
selected at random, is between 9.5
& 10.5 ohms.
Uniform pdf
The time X in hours for an engineering
student to complete a homework assignment
in ISE 261 that receives a grade above 85% is
uniformly distributed with a minimum effort of
3 hours & a maximum effort of 10 hours. What
is the probability that preparation time
exceeds 5 hours?
What is the probability that the preparation
time is within 2 hours of the mean time?
Normal Distribution
f(x;,) =
2
e-(x-) / 2
(2)
2
-< x <+
Abbreviated: X  N(,2)
Normal Distribution Interval
b
2
2
P(a<x<b) =  e-(x-) / 2 / (2) dx
a
Properties of Normal pdf’s
Bell Shaped
Symmetrical about the Mean
Unimodal
Mean = Median = Mode
One Standard Deviation is at the
inflection points on the curve
The Standard Deviation defines
the area under the pdf
Standard Normal Distribution
2
f(z;0,1) =
e-z / 2
(2)
-< z <+
Denote: N(0,1) by Z
CDF(Z) by (z)
Z Transformation
For RV X with N(,2):
Z=X-

has a standard Normal pdf
Thus
P(aXb)=P[ a-   Z  b- ]


Transformation Example
Find the probability that RV X
is greater than 5 if X is N(3,4).
Normal Distribution Example
The time to wear out of a cutting
tool edge is distributed normally
with  = 2.8 hours &  = .60 hour.
What is the probability that the tool
will wear out in less than 1.5 hour?
How often should the cutting edges
be replaced to keep the failure rate
less than 10% of the tool’s?
Normal Distribution
You are an engineering manager of a PCB
operation where the mean thickness of a
board should be 65mm with a SD of 2mm.
Previous studies indicate that this operation
can be assumed to be normally distributed.
One of the boards inspected has just
produced a thickness of 68mm. What is the
probability that a board greater than this
thickness can be produced from this
process?
Normal Distribution Example
The achievement scores for a
college entrance exam are
normally distributed with a mean
equal to 75 & standard deviation
equal to 10. What fraction of the
scores are expected to lie
between 70 & 90?
Normal Distribution Example
The width of a slot on a duralumin
forging is normally distributed with
mean equal to 0.90 inch & SD equal
to 0.002 inch. The specification
limits are given as 0.90 +/- 0.005
inch.
What percentage of forgings will be
defective?
Normal Distribution Example
A process manufactures ball
bearings whose diameters (in
2
cm) are N(2.505, 0.008 ).
Specifications call for the
diameter to be in the interval 2.5
+/- 0.01 cm. What proportion of
the ball bearings will meet the
specification?
Normal Distribution
A remote computer sends a 1 or 0 by sending either p(t) or –p(t),
respectively… where the pulse p(t) is given by:
p(t) = 1, 0  t  T
0, otherwise
When the pulse is received it is corrupted by Gaussian noise
which is added to the signal. Assume that this noise has zero
mean and variance of 1. Suppose that you wish to measure a
single value within a received pulse and use that to determine if a
1 or 0 was sent. Then the value that you measure would be a
Gaussian RV Z modeled as follows:
Z=±1+V
Where V is a Gaussian RV with zero mean and variance of 1. You
decide that a 1 was sent if the measured Z is such that Z > 0 and
otherwise will decide that a 0 was sent. Consider the case where
a 1 was sent (thus Z = +1 +V). What is the probability of making
an error?
Gamma Probability Distribution
For RV X  0:
f(x; , ) = x -1 e-x/

 ()
Where  > 0,  > 0.
Gamma Function
Definition:
() = 

-1
-x
x e dx
0
For  > 0.
Properties of the Gamma Function
(1) = 1
(1/2) =  
For any positive integer n:
(n) = (n-1)!
For  >1:
() = (-1) (-1)
Gamma pdf Properties
Standard Gamma pdf when  =1
(Incomplete Gamma function)
E(X) = 
V(X) = 2
Exponential pdf when  = 1
Chi-Square pdf when  = v/2 & 
Gamma Transformation
P(X  x) = F(x , , ) = F(x/, )
x
F(x/, ) = y -1 e-y / () dy
0
x>0
Gamma pdf Example
The chemical reaction time (in
seconds) of a commercial grade
of epoxy can be represented
with a standard Gamma pdf with
parameter  =5.0 seconds.
What is the probability that the
reaction time will be more than
4 seconds?
Gamma Distribution
In simple materials, all molecules are the
same and have the same molecular weight.
Because polymers contain molecules of
different sizes, engineers must use
probability distributions to express
molecular weight.
A commercial grade of polymer has a
molecular weight with a standard Gamma
distribution with  = 2 molar mass. What is
the probability that your randomly selected
sample has a molar mass less than 3?
Gamma pdf Example
The arrival time of large comets into
our solar system has followed a
Gamma distribution with parameters
 = 4 years &  = 15.
What is the expected arrival time of
large comets?
What is the variance in arrival time?
What is the probability of a visit by a
large comet between the years
2024 & 2039?
Gamma Distribution
A computer firm introduces a new
home computer. Past experience
indicates that the time of peak demand
(in months) after its introduction,
follows a Gamma pdf with
Variance = 36.0.
If the expected value is 18 months,
find  and .
What is the probability that the peak
demand is less than 7 months?
Exponential Probability Distribution
f(x; ) = 
Where  > 0.
-x
e
x0
Exponential pdf Properties
E(X) =  = 1 / 
2
2
V(X) =  = 1 / 
=1/
pdf easily integrated:
CDF = 
0
x
-t
e dt
=1 –
-x
e
Example Exponential pdf
A radioactive mass emits particles
according to a Poisson process at a
mean rate of 15 per minute. At
some point, a clock is started. What
is the probability that more than 5
seconds will elapse before the next
emission?
What is the mean waiting time until
the next particle is emitted?
Exponential pdf Lack of Memory Property
The lifetime of an electrical component
has an Exponential pdf with mean = 2
years. What is the probability that the
component lasts longer than 3 years?
Assume the component is now 4 years
old, and is still functioning. What is the
probability that it functions for more
than 3 additional years?
Exponential Example
Some types of equipment have
been observed to fail according to
an Exponential pdf. The time to
failure T for these types has a mean
time to failure (MTTF) equal to 1/.
The mean time to failure of a light
bulb is 100 hours, what is the
probability that a light bulb will last
for more than 150 hours?
Exponential Distribution
A computer component has a constant
failure rate of  = .02/hour. The time to
failure follows an Exponential
probability distribution. What is the
probability that the component will fail
during the first 10 hours of operation?
Suppose that the device has been
successfully operated for 100 hours.
What is the probability that it will fail
during the next 10 hours of operation?
Exponential pdf Example
A National Guard unit is supplied with
20,000 rounds of ammunition for a new
model rifle. After 5 years, 18,200 rounds
remain unused. From these,
200 rounds are chosen randomly and
test-fired. Twelve of them misfire.
Assuming that the misfires are random
failures of the ammunition caused by
storage condition, estimate the MTTF.
Chi-Squared Distribution
f(x; v) =
2
=
(v/2)-1
x
-x/2
e
2v/2 (v/2)
x0
v = number degrees of freedom
Properties:
Gamma pdf with  = v/2 &  = 2
E(2) = v
V(2) = 2v
Chi-Square pdf Example
For a Chi-Square probability
distribution with 10 degrees of
freedom, what is the
probability that the RV X is
greater than 18.307?
Chi-Square pdf Example
An unmanned space probe is directed to land
on Mars. The lateral and forward distances of
the point of impact from the target location
are independent normally distributed RVs
with zero mean and variance 5. The squared
distance between the landing point and the
target value, divided by the variance, has a
Chi-Square distribution with v = 2.
What is the probability that the distance
between the measured point and the desired
location is less than 4.80 units?
Chi-Square pdf Example
Most galaxies take the form of a flattened disc with the major
part of the light coming from this thin fundamental plane. The
degree of flattening differs from galaxy to galaxy. In the Milky
Way Galaxy most gases are concentrated near the center of
the fundamental plane. Let X denote the perpendicular
distance from this center to a random gaseous mass. RV X is
normally distributed with zero mean &  = 100 parsecs. ( A
parsec is equal to approximately 19.2 trillion miles.) The
squared distance between a scientifically measured location
of a gaseous mass and its actual location, divided by the
variance, has a Chi-Square distribution with v = 3. What is the
probability that the distance between the measured point and
the actual location is greater than 144.25 parsecs?
Weibull Distribution
f(x; , ) =
 x -1 e-(x/)



Parameters:  > 0 &  >0
x0
Weibull pdf Properties
Exponential when  = 1
(constant failure rate)
E(X)=  (1+1/)
V(X)= 2{(1+2/)–[(1+1/)]2}

CDF= 1 – e-(x/)
For x  0
Weibull pdf Example
The lifetime measured in years
of a brand of TV picture tubes
follows a Weibull probability
distribution with parameters
 = 2 years &  = 13. What is the
probability that a TV tube fails
before the expiration of the
two-year warranty?
Weibull pdf Example
The bake step in the manufacture of a
semiconductor follows a Weibull pdf with
 = 2.0 hours and  = 10. What is the
expected amount of time needed to bake a
semiconductor?
What is the probability that a randomly
chosen bake step takes longer than 4 hours?
What is the probability that it takes between
2 & 7 hours?
Weibull Threshold Example
An extremely hard silicon carbide
composite has a tensile strength in
MegaPascals that under specified
conditions can be modeled by a Weibull
distribution after a minimum tensile
strength of 2 MPa with  = 9 and  =
180. What strength value separates the
weakest 10% of all specimens from the
remaining 90%?
Weibull Distribution
An electronic device has decreasing
failure rate characterized by the two
parameter Weibull pdf with  = 0.50 & 
= 180 years. The device is required to
exhibit a design life reliability of 0.90.
What is the design life if there is no
wear-in period?
What is the design life if the device is
first subject to a wear-in period of one
month? (Hint: conditional probability)
Lognormal Distribution
f(x; , ) =
2
2
-[ln(x)

]
/
(2)
e
2 x
For x  0
Parameters ln & ln of ln(x)
Properties of Lognormal pdf
ln(x) is a Normal pdf
E(X) = e
2
 +  /2
2
2
V(X) = e 2 +  (e  - 1)
CDF = [(ln(x) - ) / ]
note:  is CDF of Z
If RVs X & Y are LN than XY is LN
Lognormal pdf Example
The lifetime of a certain insect is
Lognormally distributed with
parameters ln = 1 week & ln = ½
weeks. What is the expected
lifetime of a randomly selected
insect?
What is the standard deviation of
the lifetime?
What is the probability that a insect
lives longer then 4 weeks?
Lognormal Example
Fatigue life data for an industrial
rocker arm is fit to a Lognormal
pdf. The following parameters are
obtained: ln = 16.8 cycles &
ln = 2.3. To what value should
the design life be set if the
probability of failure is not to
exceed 1.0%?
Beta Probability Distribution
f(x; , , A, B)
=
1
B-A
(+)
()()
-1
x-A
B-A
-1
B-x
B-A
For x on an interval A  x  B.
Parameters  > 0 &  >0.
Properties of Beta pdf
E(X) = A + (B - A)
V(X) =

( + )
(B - A)2 
(+)2(++1)
A = 0 & B = 1 yields standard Beta
Beta pdf Example
The proportion X of a surface area in a
randomly selected forest is covered by Lilac
trees. The RV X has a standard Beta
distribution with  = 5 and  = 2. What is the
expected proportion of the surface area to be
covered by this tree?
The first quadrant is taken for study. What is
the probability that the proportion of surface
area to be covered by this tree is less than or
equal to 20%?
Beta pdf Example
Time analysis in the construction industry
assumes that the time necessary to
complete any particular activity once it has
been initiated has a Beta pdf with A = the
optimistic time (no mistakes are made) and
B = the pessimistic time (all possible
mistakes are made). If the time T (days) for
laying the foundation of a single family
home has a Beta pdf with A = 2, B = 5,  =2,
&  =3, what is the probability that it takes at
most 3 days to lay the foundation?