1 Continuous Probability Distributions Continuous Random Variables & Probability Distributions Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
25
Embed
Continuous Probability Distributions Continuous Random Variables & Probability Distributions
Systems Engineering Program. Department of Engineering Management, Information and Systems. EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS. Continuous Probability Distributions Continuous Random Variables & Probability Distributions. - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Continuous Probability Distributions
Continuous Random Variables &Probability Distributions
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
Systems Engineering ProgramDepartment of Engineering Management, Information and Systems
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
2
•Definition - A random variable is a mathematical function that associates a number with every possible outcome in the sample space S.
• Definition - If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space and a random variable defined over this space is called a continuous random variable.
• Notation - Capital letters, usually or , are used to denote random variables. Corresponding lower case letters, x or y, are used to denote particular values of the random variables or .
X Y
YX
Random Variable
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
3
For many continuous random variables or (probabilityfunctions) there exists a function f, defined for allreal numbers x, from which P(A) can for any eventA S, be obtained by integration:
Given a probability function P() which may berepresented in the form of
A
dxxfAP
areaA
dxxfAP
Continuous Random Variable
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
4
in terms of some function f, the function f is calledthe probability density function of the probabilityfunction P or of the random variable , and the probability function P is specified by the probability density function f.
X
Continuous Random Variable
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
5
Probabilities of various events may be obtained from the probability density function as follows:
Let A = {x|a < x < b}
Then
P(A) = P(a < X < b)
A
dxxf
b
a
dxxf
Continuous Random Variable
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
6
Therefore = area under the density function curvebetween x = a and x = b.
f(x)
x
Area = P(a < x <b)
a b0 0
)(AP
Continuous Random Variable
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
7
The function f(x) is a probability density function for the continuous random variable X, defined over the set of real numbers R, if
1. f(x) 0 for all x R.
2.
3. P(a < X < b) =
.1dx)x(f
b
a
.dx)x(f
Probability Density Function
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
8
The cumulative probability distribution function, F(x), of a continuous random variable with density function f(x) is given by
Note:
x
.dt)t(f)xX(P)x(F
xFdx
df(x)
X
Probability Distribution Function
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
9
Probability Density and Distribution Functions
f(x) = Probability Density Function
x
Area = P(x1 < <x2)
F(x) = Probability Distribution Function
x
F(x2)
F(x1)
x2x1
P(x1 < <x2) = F(x2) - F(x1)
1
cumulative area
x2x1
X
X
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
10
• Mean or Expected Value
• Remark
Interpretation of the mean or expected value:The average value of in the long run.
dxxfx XEμ
X
Mean & Standard Deviationof a Continuous Random Variable X
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
11
•Variance of X:
•
•Standard Deviation of :
dx f(x) μ)-(xσXVar 22
22 μXEXVar
22 μxfx dx
XVarσ X
Mean & Standard Deviationof a Continuous Random Variable X
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
12
If a and b are constants and if = E is the meanand 2 = Var is the variance of the randomvariable , respectively, then
and
baμbaXE
XVarabaXVar 2
X)(X
)(X
Rules
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
13
If Y = g(X) is a function of a continuous random variable , then