STT 430, Summer 2006
Lecture 3
Materials Covered: Chapter 4
Suggested Exercises: 4.1, 4.3, 4.4, 4.6, 4.11, 4.15, 4.18, 4.32, 4.46, 4.47, 4.57, 4.64, 4.77, 4.117.
1. Cumulative Distribution Function.

Definition 4.1: Let Y denote any random variable. The cumulative distribution function
(CDF) of Y, denoted by F(y), is given by F(y) = P(Yy) for -<y<.
Example 4.1: Suppose that Y has binomial distribution with n=2 and p=1/2. Find F(y).
Distribution functions for discrete random variable are always step functions because the CDF
increases only at a countable number of points.

Properties of a CDF.
Theorem 4.1: If F(y) is a CDF, then
(1). F ()  lim F ( y)  0.
y 
(2). F ()  lim F ( y)  1.
y 
(3). F ( y ) is nondecreasing function of y.
2. Continuous Random Variable and it’s Probability Distribution.

Definition 4.2: Let Y denote a random variable with CDF F(y). Y is said to be continuous
if the distribution function F(y) is continuous for    y   .

Definition 4.3: Let F(y) be the CDF for a continuous random variable Y. The f(y), given by
f ( y) 
dF ( y )
 F ( y )
dy
wherever the derivative exists, is called the probability density function (PDF) of the random
variable Y.
STT 430, Summer 2006
Therefore, F(y) can be written as
y
F ( y)   f (t )dt

Graphically, we have:

Properties of a PDF.
Theorem 4.2: If f(y) is a PDF, then
(1). f(y)  0 for any value of y.
(2).



f ( y)dy  1 .
Example 4.2: Suppose that
 0, for y  0

F ( y )   y, for 0  y  1
 1, for y  1

Find the PDF of Y, and graph it.
Example 4.3: Let Y be a continuous random variable with PDF given by
3 y 2 , 0  y  1
f ( y)  
 0, otherwise
Find F(y). Graph both f(y) and F(y).
STT 430, Summer 2006

Theorem 4.3: If the random variable Y has density function f(y) and ab, then the
probability that Y falls in the interval [a,b] is
b
P(a  Y  b)   f ( y)dy.
a
Example 4.4: Given f ( y)  cy ,0  y  2, and f(y)=0 elsewhere, find the value of c for
2
which f(y) is a valid density function.
Example 4.5: Find P(1Y2) for Example 4.4. Also find P(1<Y<2).
3. Expected Values for Continuous Random Variable.
Definition 4.4: The expected value of a continuous random variable Y is

E (Y )   yf ( y)dy

provided that the integral exists.
Theorem 4.4: Let g(Y) be a function of Y; then the expected value of g(Y) is given by

E[ g (Y )]   g ( y) f ( y)dy

provided that the integral exists.
STT 430, Summer 2006
Theorem 4.5: Let c be a constant, and let g1(Y), g2(Y), … , gk(Y) be functions of a continuous
random variable Y. Then the following results hold:
(1). E(c) = c.
(2). E[cg(Y)] = c E[g(Y)].
(3). E[g1(Y)+g2(Y)+…+gk(Y)] = E[g1(Y)]+E[g2(Y)]+…+E[gk(Y)].
Example 4.6: In example 4.4 we determined that f(y)=(3/8)y2 for 0y2, f(y)=0 elsewhere, is a
valid density function. If the random variable Y has this density function, find E(Y) and V(Y).
4. Examples of Continuous Random Variables.
(1).The Uniform Probability Distribution.
Definition 4.5: If
1   2 , a random variable Y is said to have a uniform probability
distribution on the interval ( 1 , 2 ), denoted by Y ~ U( 1 , 2 ), if and only
if the density function of Y is
 1
  
f ( y)   2 1

 0
1  y   2
.
otherwise
Theorem 4.6: If Y ~ U( 1 , 2 ), then
E (Y ) 
Proof:
1   2
2
,
and
V (Y ) 
( 2  1 ) 2
.
12
STT 430, Summer 2006
Example 4.7: It is known that, during a given 30-minute period, one customer will arrive at a
checkout counter any time within the 30-minute period. Find the probability that the customer will
arrive during the last 5 min of the 30-minute period.
(2). The Normal Probability Distribution.
Definition 4.7: A random variable Y is said to have a normal probability distribution, denoted
by Y~N(, ) if and only if the density function of Y is given by

1
f ( y) 
e
 2
( y  )2
2 2
   y  ,
,
where -<<, and  > 0.
Theorem 4.7: If Y ~ N(, ), then
E (Y )   ,
V (Y )   2 .
and
Example 4.7: Let Z denote a normal random variable with mean 0 and standard deviation 1.
a. Find P(Z >2).
b. Find P(-2Z2).
c. Find P(0Z1.73).
Theorem: If Y ~ N(, ), then
Z
Y 

~ N (0,1) .
STT 430, Summer 2006
Example 4.9: The achievement score for a college entrance examination are normally
distributed with mean 75 and standard deviation 10. What fraction of the scores lies between
80 and 90?
(3). The Gamma Probability Distribution.
Definition 4.8: A random variable Y is said to have a Gamma probability distribution with
parameters >0 and >0 if and only if the density function of Y is given by
 y  1e  y 

f ( y )     ( )

0
where ( ) 


0
y0
,
elsewhere
y  1e  y dy.
Theorem 4.8: If Y has a gamma distribution with parameter  and  , then
E (Y )   ,
Proof:
and
V (Y )   2 .
STT 430, Summer 2006
Some Special Gamma Distributions.
(i)
Chi-square probability distribution: Let v be a positive integer. A random variable Y
is said to have a chi-square distribution with v degrees of freedom if and only if Y has a
gamma distribution with parameter  = v/2 and  = 2.
Easy to see that if Y is a chi-square random variable with v degrees of freedom, then
E(Y) =v, V(Y) = 2v.
(ii)
Exponential probability distribution: A random variable Y is said to have an
exponential distribution with parameter  > 0 if and only if Y has a gamma distribution
with parameter  =1 and >0.
Easy to see, the density function of Y is given by
 1 y 
 e
f ( y)   

 0
y0
,
elsewhere
and E(Y) = , V(Y) = 2.
Example 4.10: Suppose that Y has an exponential probability density function. Show that, if a>0
and b>0, then
P(Y > a + b | Y > a) = P(Y > b).
(4). The Beta Probability Distribution (Omit).
5. Tchebysheff’s Theorem
Theorem 4.13: Let Y be a random variable with finite mean  and variance 2. Then for any k>0,
P(| Y   | k )  1 
Proof:
1
1
, or P (| Y   | k )  2 .
2
k
k
STT 430, Summer 2006
Example 4.17: Suppose that experience has shown that the length of time Y (in minutes) required
to conduct a periodic maintenance check on a dictating machine follows a gamma distribution
with  = 3.1 and  = 2. A new maintenance worker takes 22.5 min to check the machine. Does
this length of time to perform a maintenance check disagree with prior experience?
6. Other Expected Values of Continuous Variables.
(1). k-th moment about the origin.
(2). k-th moment about the mean.
(3). Moment generating function.