# Lecture Note 8 ```ST361: Ch5.4 + Ch1.3 Random Variable and Its Probability Distribution
----------------------------------------------------------------------------------------------------------Topics: Random Variable (&sect;5.4)
Probability Distribution of a discrete random variable (&sect;5.4, &sect;1.3)
Mean and Variance of a discrete random variable (&sect;5.4)
Probability Distribution of a continuous variable (&sect;5.4, &sect;1.3)
Mean and Variance of a continuous random variable (&sect;5.4)
---------------------------------------------------------------------------------------------------------- Random Variable (r.v.)
 A random variable is a variable whose value is a __________________
_________________________________ of an experiment.
 We can think that an r.v. is any rule that associates a _______________
with each ____________ in an experiment.
Ex. Consider an experiment of tossing 2 coins. One way to define a r.v. is

For numerical variables, most of the time the values themselves can be
used as a r.v.
Ex. Define a r.v. to for the exam score of a student as
----------------------------------------------------------------------------------------------------- Discrete r.v.
The possible values of the r.v. are isolated points along the number line.
Ex. x = # of Heads of tossing 2 coins x, =0,1,2
Ex. x = # of telephone lines in a company that is in use, x =0,1,2, 3,……
(c.f. Continuous r.v.: The possible values forms an interval along the real line)
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 Probability Distribution of a Discrete r.v.
1. The probability distribution of a r.v., denoted as _________, describes
(a) ________________________________________________ and
(b) _________________________________________________
Ex. X = result of tossing a fair dice. The probability distribution of x is
2. In general, the probability that x gets a value c, P(x=c), is defined as the
sum of all corresponding outcomes in S (i.e., the sample space) that
are assigned to the value x.
Ex. X = # of heads in tossing two fair coins. Then the probability
distribution of X is
3. There are 3 ways to display a probability distribution for a discrete r.v.:
Ex. Toss a (unfair) coin 3 times, and let x= # of heads. Then the probability
distribution of x is given as below:
(1) Density plot
2
(2) Table
x
P(x)
0
0.1
1
0.4
2
0.3
3
0.2
(3) Formula
 From the probability distribution, we can calculate
P( x = 3 ) =
P( x &lt; 2 ) =
P( x  2 ) =
P( x &gt; 0 ) =
4. For any probability distribution P(x), (recall the axiom of probabilities…)
(1)
(2)
Ex. (1) Find the value of c so that the following function is a probability
distribution of a r.v. x: P  x   c  x  2 , where x  0,1,2,3
(2) For this probability distribution, find P(x  2)
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 FYI: Mean and Variance of a discrete r.v. with probability distribution P  x 
The mean  x   x  P  x 
x
(The mean of a r.v. is also called as expected value.)
The variances  x2    x   x   P  x 
2
x
The standard deviation  =
Ex. A contractor is required by a county planning department to submit from 1 to 5
different forms, depending on the nature of the project. Let x = # of forms required of
the next contractor, and px  kx for x=1,2,3,4,5.
(a) What is the value of k?
(b) What is the probability that at most 3 forms are required?
(c) What is the expected number (i.e., mean) of forms required?
(d) What is the SD of the number of forms required? (This calculation won’t be included in
the exams)
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 Continuous Random Variable (r.v.)
A r.v. is continuous if its possible values forms an interval along the real line
Ex. x = exam score, 0  x  100
Ex. x = your height, 0  x  
 Probability Distribution of a continuous r.v.
1. Every continuous r.v. x has a __________________________, denoted
as ________ such that for any 2 numbers a and b (a&lt;b),
P( a  x  b ) = ________ under the density curve of f(x) between a and b
Ex. P( -1  X  1 ) =
Ex. P( X  1 ) =
Comment: For continuous r.v. x,
(1) Probability is the area encompassed by the density curve, the two
vertical bars and the x-axis. _______________________________
______________________________________________________
(2) Because area under the curve represents probabilities, the total area
under the density curve should be equal to __________
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(3) Unlike the discrete r.v., the Y-axis is not probability
( The height is determined so that ________________
__________________________________________
Ex.
?
(4) P( X = a ) = _________ (Think what is the size of the corresponding area?)
(5) For a continuous r.v.,
(6) Ways to presenting the density function of a continuous r.v.: by a
density plot or formula
(see next page)
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Ex. Consider a r.v. X= test score. The probability distribution of X is
given below.
1) Density plot
2) Density function
if x  40
 ________
 1
1

y if

 30 1200
2

if
f x   
120
3

if

120


if x  100
 ______
P( X=70) =
P( 60 &lt; X &lt; 80 ) =
P(X &gt; 70 ) =
P( X  70 ) =
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Ex.
P X  20 
P X  20 
P5  X  15 
Summary: A density function f  x  of a r.v. has to satisfied the following properties:
(a) f  x   0
THINK: do we need 0  f  x   1 ?
(b) The total area under the curve is 1, i.e.,



f  x  dx  1
Why?
 FYI: Mean and Variance of a continuous r.v. with density function f  x 

The mean x   x  f  x  dx

(The mean of a r.v. is also called as expected value.)
The variances  2  


 x  x 
The standard deviation  
2
 f  x  dx

 x   

x
2
 f  x  dx
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