Matakuliah
Tahun
Versi
: I0284 - Statistika
: 2008
: Revisi
Pertemuan 07
Variabel Acak Diskrit dan Kontinu
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menghitung
Peluang, nilai harapan, dan varians
variabel acak diskrit dan kontinu.
2
Outline Materi
•
•
•
•
Definisi variabel acak
Distribusi probabilitas diskrit
Distribusi probabilitas kontinu
Nilai harapan dan varians
3
Random Variable
• Random Variable
– Outcomes of an experiment expressed
numerically
– E.g., Toss a die twice; count the number of
times the number 4 appears (0, 1 or 2 times)
– E.g., Toss a coin; assign $10 to head and $30 to a tail
= $10
T
= -$30
4
Discrete Random Variable
• Discrete Random Variable
– Obtained by counting (0, 1, 2, 3, etc.)
– Usually a finite number of different values
– E.g., Toss a coin 5 times; count the number of
tails (0, 1, 2, 3, 4, or 5 times)
5
Discrete Probability
Distribution Example
Event: Toss 2 Coins
Count # Tails
Probability Distribution
Values
Probability
T
T
T
T
0
1/4 = .25
1
2/4 = .50
2
1/4 = .25
This is using the A Priori Classical
Probability approach.
6
Discrete Probability
Distribution
• List of All Possible [Xj , P(Xj) ] Pairs
– Xj = Value of random variable
– P(Xj) = Probability associated with value
• Mutually Exclusive (Nothing in Common)
• Collective Exhaustive (Nothing Left Out)
0  PX j  1
PX  1
j
7
Summary Measures
• Expected Value (The Mean)
– Weighted average of the probability
distribution
–
  E  X    X jP X j 
j
– E.g., Toss 2 coins, count the number of tails,
compute expected value:
   X jP  X j 
j
  0 .25  1.5   2 .25  1
8
Summary Measures
(continued)
• Variance
– Weighted average squared deviation about the
mean
2
– 2  E X  2 
X  P X





j
  
j
– E.g., Toss 2 coins, count number of tails,
compute variance:
   X j    P  X j 
2
2
  0  1 .25  1  1 .5   2  1 .25
2
 .5
2
2
9
Computing the Mean for
Investment Returns
Return per $1,000 for two types of investments
P(Xi) P(Yi)
Investment
Economic Condition Dow Jones Fund X Growth Stock Y
.2
.2
Recession
-$100
-$200
.5
.5
Stable Economy
+ 100
+ 50
.3
.3
Expanding Economy
+ 250
+ 350
E  X    X   100.2  100.5   250.3  $105
E Y   Y   200.2  50.5  350 .3  $90
10
Computing the Variance for
Investment Returns
P(Xi) P(Yi)
Investment
Economic Condition Dow Jones Fund X Growth Stock Y
.2
.2
Recession
-$100
-$200
.5
.5
Stable Economy
+ 100
+ 50
.3
.3
Expanding Economy
+ 250
+ 350
  .2  100  105   .5 100  105   .3 250  105 
2
2
X
2
 X  121.35
 14, 725
  .2  200  90   .5  50  90   .3 350  90 
2
2
Y
 37,900
2
2
 Y  194.68
11
2
Continuous Random Variables
A random variable X is continuous if its
set of possible values is an entire
interval of numbers (If A < B, then any
number x between A and B is possible).
12
Probability Density Function
For f (x) to be a pdf
1. f (x) > 0 for all values of x.
2.The area of the region between the
graph of f and the x – axis is equal to 1.
y  f ( x)
Area = 1
13
Probability Distribution
Let X be a continuous rv. Then a
probability distribution or probability
density function (pdf) of X is a function
f (x) such that for any two numbers a
and b,
P  a  X  b    f ( x)dx
b
a
The graph of f is the density curve.
14
Probability Density Function
P(a  X  b) is given by the area of the shaded
region.
y  f ( x)
a
b
15
Important difference of pmf and pdf
Y, a discrete r.v. with pmf f(y)
X, a continuous r.v. with pdf f(x);
• f(y)=P(Y = k) = probability that the outcome is k.
• f(x) is a particular
 function with the property that
for any event A (a,b), P(A) is the integral of f
over A.
b
P( A)   f ( x)dx   f ( x)dx
A
a
k
P( X  k )   f ( x)dx  0
k
16
Ex 1. (4.1) X = amount of time for which a book
on 2-hour reserve at a college library is checked
out by a randomly selected student and suppose
that X has density function.
0.5 x 0  x  2
f ( x)  
otherwise
0
1 21
f ( x)dx   0.5 xdx  x 0.25
0
4 0
1
1
a. P ( x  1)  

1.5
b. P (0.5  x  1.5)   0.5 xdx 0.5
0. 5
c. P x  1.5   0.5 xdx 0.4375
2
1.5
17
Probability for a Continuous rv
If X is a continuous rv, then for any
number c, P(x = c) = 0. For any two
numbers a and b with a < b,
P ( a  X  b)  P ( a  X  b)
 P ( a  X  b)
 P ( a  X  b)
18
Expected Value
• The expected or mean value of a continuous rv X
with pdf f (x) is
X  E  X  

 x  f ( x)dx

• The expected or mean value of a discrete rv X
with pmf f (x) is
E( X )   X 
 x  p ( x)
xD
19
Variance and Standard Deviation
The variance of continuous rv X with
pdf f(x) and mean  is
2
X

 V ( x) 
 (x  )

2
 f ( x)dx
 E[ X    ]
2
The standard deviation is  X  V ( x).
20
Short-cut Formula for Variance
    E ( X )
V (X )  E X
2
2
21
• Selamat Belajar Semoga Sukses.
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