Home
Add Document
Sign In
Create An Account
Continuous Random Variables
Continuous Random Variables Daniel A. Menascé, Ph.D. Department of Computer Science George Mason University 1 © 2001 D. A. Menascé. All Rights Reserve...
Author:
Calvin Barker
3 downloads
1 Views
36KB Size
Report
Download PDF
Recommend Documents
Continuous Random Variables
5 Continuous random variables
Continuous Random Variables
Continuous Random Variables
Continuous Random Variables
Continuous Random Variables
4.5 Continuous Random Variables
Continuous Random Variables
Continuous Random Variables
Continuous Random Variables
Continuous Random Variables
Chapter5 Continuous Random Variables
Continuous Random Variables: Introduction
Lab 3. Continuous Random Variables
Continuous Random Variables and Distributions
Chapter 4 Continuous Random Variables
6 Jointly continuous random variables
6 Jointly continuous random variables
Working with Continuous Random Variables
Chapter 6: Continuous Random Variables
Lecture 03: Continuous random variables
Continuous Random Variables & Probability Distributions
Continuous Random Variables Lecture 4
Probability Models.S3 Continuous Random Variables
Continuous Random Variables Daniel A. Menascé, Ph.D. Department of Computer Science George Mason University 1 © 2001 D. A. Menascé. All Rights Reserved.
Relevant Functions • Probability density function (pdf) of r.v. X: f X (x ) b
P[a ≤ X ≤ b] = ∫ f X ( x)dx a
• Cumulative distribution function (CDF):
F X ( x) = P[ X ≤ x] • Tail of the distribution (reliability function):
R X ( x) = P[ X > x] = 1 − FX ( x) 2 © 2001 D. A. Menascé. All Rights Reserved.
1
Moments +∞
• k-th moment: E[ X k ] = ∫− ∞ x k f X ( x)dx • Expected value (mean): first moment µ = E[ X ] = ∫
+∞
−∞
xf X ( x) dx
• k-th central moment: E[( X − µ )k ] = ∫
+∞
−∞
( x − µ ) k f X (x )dx
• Variance: second central moment σ 2 = E [( X − µ )2 ] = ∫
+∞
−∞
( x − µ )2 f X ( x )dx
3 © 2001 D. A. Menascé. All Rights Reserved.
The Uniform Distribution • pdf:
1 a ≤ x≤ b f X ( x) = b − a 0 otherwise
• Mean:
µ=
a +b 2
• Variance: σ 2 = (b − a )
2
12
4 © 2001 D. A. Menascé. All Rights Reserved.
2
The Uniform Distribution 1
U(0,1)
0
0.2
0.5
1
P[0.2τ ] ~ P[ t ≤ t + τ ] − P[ ~ t ≤τ ] = ~ P[ t > τ ] 1 − e− λ.(t +τ ) − (1 − e −λ.τ ) = 1 − (1 − e −λ .τ ) = 1 − e −λ .t 19 © 2001 D. A. Menascé. All Rights Reserved.
Exponential Distribution 1.0
0.9
0.8
In Excel: FX(x) = EXPONDIST(x,λ,TRUE) fX(x) = EXPONDIST(x,λ,FALSE)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 0
2
4
6 pdf
8
10
12
CDF
20 © 2001 D. A. Menascé. All Rights Reserved.
10
Pareto Distribution • A case of a heavy-tailed distribution. • The probability of large values is not negligible. a
f X (x ) =
a>0 ,
1 +a
x
1 a x a >1
FX ( x) = 1 −
• Mean:
a a −1
a>0 ,
a (a −1) ( a − 2)
• Variance:
x ≥1 x ≥1
a>2
2
21 © 2001 D. A. Menascé. All Rights Reserved.
Tail of the Pareto and Exponential Distributions Ln x 0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
-10
-20
-30
-40
-50
Ln P[X>x] -60
Tail Pareto
Tail Exponential
22 © 2001 D. A. Menascé. All Rights Reserved.
11
Generation of Random Variables 1.0
• randomly generate a number u = U(01,) • x = F-1 (u) where F is the CDF
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 -10 -9
-8
-7
-6
-5
-4 -3.5 -3 -2.5
-2 -1.5
-1 -0.5 0
0.5
1
1.5
2
2.5
3
3.5
4
5
6
7
8
9
10
23 © 2001 D. A. Menascé. All Rights Reserved.
12
Suggest Documents
Continuous Random Variables
Read more
5 Continuous random variables
Read more
Continuous Random Variables
Read more
Continuous Random Variables
Read more
Continuous Random Variables
Read more
Continuous Random Variables
Read more
4.5 Continuous Random Variables
Read more
Continuous Random Variables
Read more
Continuous Random Variables
Read more
Continuous Random Variables
Read more
Continuous Random Variables
Read more
Chapter5 Continuous Random Variables
Read more
Continuous Random Variables: Introduction
Read more
Lab 3. Continuous Random Variables
Read more
Continuous Random Variables and Distributions
Read more
Chapter 4 Continuous Random Variables
Read more
6 Jointly continuous random variables
Read more
6 Jointly continuous random variables
Read more
Working with Continuous Random Variables
Read more
Chapter 6: Continuous Random Variables
Read more
Lecture 03: Continuous random variables
Read more
Continuous Random Variables & Probability Distributions
Read more
Continuous Random Variables Lecture 4
Read more
Probability Models.S3 Continuous Random Variables
Read more
×
Report "Continuous Random Variables"
Your name
Email
Reason
-Select Reason-
Pornographic
Defamatory
Illegal/Unlawful
Spam
Other Terms Of Service Violation
File a copyright complaint
Description
×
Sign In
Email
Password
Remember me
Forgot password?
Sign In
Login with Google
Login with Facebook